What is the end behavior of \(f ( x )\) in the function \(f ( x )\) \(= \log ( x - 2 )\) as \(x\) approaches \(2 ?\) 

A. \(f ( x ) \rightarrow - \infty\) 

B. \(f ( x ) \rightarrow - 2\) 

C. \(f ( x ) \rightarrow 0\) 

D. \(f ( x ) \rightarrow 2\) 

E. \(f ( x ) \rightarrow \infty\) 

Consider the integral: \( \int _ { 0 } ^ { 4 } ( 4 x - x ^ { 2 } ) d x\) 

The graph of \( f ( x ) = 4 x - x ^ { 2 } \) is below: 

a) Find the exact value of the integral using Riemann sums. Use the following limit definition for area: Area \( = \lim \limits_ { n \rightarrow \infty } \sum\limits _ { i = 1 } ^ { n } f ( x _ { i } ) \Delta x\) 

You will need to use the following summation formulas: 

\(\sum \limits_ { i = 1 } ^ { n } i = \frac { n ( n + 1 ) } { 2 } \quad\quad \sum\limits _ { i = 1 } ^ { n } i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }\)

b) Find the exact value of the integral by using the Fundamental Theorem of Calculus (evaluate the definite integral using antiderivative). 

Draw a dependency diagram and write a chain rule formula for \( \frac { \partial z } { \partial q } \) and \( \frac { \partial z } { \partial r } \) given the functions below. 

\(z = k ( c , d ) , c = f ( q , r ) , d = h ( q , r )\)

Choose the correct chain rule formula for \(\dfrac{\partial z}{\partial r}\) 

A. \( \frac { \partial z } { \partial r } = \frac { \partial z } { \partial c } \frac { \partial z } { \partial d } + \frac { \partial c } { \partial r } \frac { \partial d } { \partial r }\) 

B. \( \frac { \partial z } { \partial r } = \frac { \partial z } { \partial c } \frac { \partial d } { \partial r } + \frac { \partial z } { \partial d } \frac { \partial c } { \partial r }\) 

C. \( \frac { \partial z } { \partial r } = \frac { \partial z } { \partial c } \frac { \partial c } { \partial r } + \frac { \partial z } { \partial d } \frac { \partial d } { \partial r }\) 

D. \( \frac { \partial z } { \partial r } = \frac { \partial z } { \partial c } + \frac { \partial c } { \partial r } + \frac { \partial z } { \partial d } + \frac { \partial d } { \partial r }\) 

Does the graph of this function have a horizontal asymptote? If it does, write its equation. 

\( y = \frac { 2 x ^ { 2 } - 9 } { - x ^ { 2 } + 1 }\) 

The Cross Country Bus Tour Company owns five buses that tour the United States. All tours start in Los Angeles. The cost \( ( C ) \) to operate their tours is \( \$ 250.00 \) per hour plus \( \$ 1000 \) for marketing for each trip. 

Tickets for the Wisconsin tour cost \( \$ 700 \) . How many tickets does the tour company need to sell to break even? 

A. 23

B. 25

C. 27

D. 29

The frequency at which a guitar string vibrates (which determines the pitch of the note we hear) is related to the tension \( T \) to which the string is tightened, the density \( \rho \) of the string, and the effective length \( L \) of the string by the equation \( f = \frac { 1 } { 2 L } \sqrt { \frac { T } { \rho } } \) . By running a finger along a string, a guitarist can change the distance between the bridge and their finger. Suppose that \( L = 1 / 2 m \) and \(\sqrt{\frac{T}{\rho} } = 70m/s\) so hat the units of \( f \) are Hertz (cycles per second). 

If the guitarist's hand slides so that \( L ^ { \prime } ( t ) = - 1 \) , find \( f ^ { \prime } ( t ) \) . At this rate, how long will it take to raise the pitch one octave (that is, double \( f \)) ? 

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the \( x \) -axis over the given interval. 

\( f ( x ) = 2 x + 1 , [ 0,2 ] , 4 \) rectangles 

Use the information to find and compare \( \Delta y \) and \( d y \) . (Round your answers to three decimal places.) 

\(y = 0.7 x ^ { 9 } \quad x = 1 \quad \Delta x = d x = 0.1\)

\(\Delta y = \)  _______

\( d y = \) _______

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the \( x \) -axis over the given interval. 

\( f ( x ) = 2 x + 1 , [ 0,2 ] , 4 \) rectangles 

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the \( x \) -axis over the given interval. (Round your answers to four decimal places.) 

\( g ( x ) = 3 \sin x , [ 0 , \pi ] , 6 \) rectangles 

left endpoints approximation Area: _______

right endpoints approximation Area: _______

Use the Midpoint Rule with \( n = 4 \) to approximate the area of the region bounded by the graph of the function and the \( x \) -axis over the given interval. 

\( f ( x ) = x ^ { 2 } + 5 , [ 0,2 ]\) 

Use the Midpoint Rule with \( n = 4 \) to approximate the area of the region bounded by the graph of the function and the \( x \) -axis over the given interval. (Round your answer to three decimal places.) 

\( f ( x ) = 5 \tan x , [ 0 , \frac { \pi } { 4 } ]\) 

If an object moves along the \( y \) -axis (marked in feet) so that its position at time \( x \) (in seconds) is given by \( f ( x ) = 182 x - 13 x ^ { 2 } \) , find the following. 

(A) The instantaneous velocity function \( v = f ^ { \prime } ( x ) \) 

(B) The velocity when \( x = 0 \) and \( x = 4 \) sec 

(C) The time(s) when \( v = 0 \) 

First use the appropriate properties of logarithms to rewrite \( f ( x ) \) , and then find \( f ^ { \prime } ( x )\) 

\(f ( x ) = 23 x + \ln 23 x\)

Use the Midpoint Rule with \( n = 4 \) to approximate the area of the region bounded by the graph of the function and the \( x \) -axis over the given interval. (Round your answer to three decimal places.) 

\( f ( x ) = 5 \tan x , [ 0 , \frac { \pi } { 4 } ]\) 

Find the general solution of the differential equation and check the result by differentiation. (Use \( C \) for the constant of integration.) 

\( \frac { d y } { d x } = 6 x ^ { 5 / 6 } \) 

A point is moving on the graph of \( x y = 21 \) . When the point is at \( ( 7,3 ) \) , its \( x \) -coordinate is increasing by \( 7 \) units per second. How fast is the \( y \) -coordinate changing at that moment? 

Calculate the definite integral by refering to the figure with the indicated areas. 

Find \(\int _ { a } ^ { c } f ( x ) d x\) 

Area \( A = 1.335 \)      Area \( C = 5.869 \) 

Area \( B = 2.387 \)      Area \( D = 1.626 \) 

Find the critical values \( \chi _ { R } ^ { 2 } \) and \( \chi _ { L } ^ { 2 } \) for the given confidence level \( c \) and sample size \( n \) , also find the population variance and population standard deviation. 

\(c = 0.98 , \quad n = 29 , s ^ { 2 } = 1.44\)

Follow the steps below to find the nonnegative numbers \( x \) and \( y \) that satisfy the given requirements. Give the optimum value of the indicated expression. Complete parts \( ( a ) \) through \( ( f ) \) below. 

\( x + y = 170 \) and the product \( P = xy \) as large as possible. (

b) Substitute the result from part (a) into \( P = x y \) , the equation for the variable that is to be maximized. 

\( P = x ( 170 - x ) \) (Type an equation.) 

(c) Find the domain of \( P \) found in part (b). [0,170] (Type your answer in interval notation.) (d) Find \( \frac { d P } { d x } \) . Solve the equation \( \frac { d P } { d x } = 0 \) . 

\( \frac { d P } { d x } = \square \) and when \( \frac { d P } { d x } = 0 , x = \square \) (Use a comma to separate answers as needed.) 

Consider the region enclosed by \( y = 4 x - x ^ { 2 } \) and \( x \) -axis on the first quadrant. Draw the graph on \( x - y \) plane and answer the following questions.

(a) Write an integral that will give the area of the region in terms of \( d x \) no sol given 

(b) Write the integrals (do not solve) that give the volume of the solid obtained by rotating the region 

      i. Rotate about \( x \) -axis (disc method) no sol given 

      ii. Rotate about \( y = - 2 \) (Washer method) no sol given 

      iii. Rotate about \( y \) -axis (shell method) no sol given 

      iv. Rotate about \( x = - 2 \) (shell method) no sol given 

Use the following figure, which shows a graph of \( f ( x ) \) to find each of the indicated integrals. 

Note that the first area (with vertical, red shading) is \( 33 \) and the second (with oblique, black shading) is \( 11 \) . 

A. \( \int _ { a } ^ { b } f ( x ) d x = \) _______

B. \( \int _ { b } ^ { c } f ( x ) d x = \) _______

C. \( \int _ { a } ^ { c } f ( x ) d x = \) _______

D. \( \int _ { a } ^ { c } | f ( x ) | d x = \) _______

Let \( y ( t ) \) represent your bank account balance, in dollars, after \( t \) years. Suppose you start with 

\( \$ 50000 \) in the account. Each year the account earns \( 5 \% \) interest, and you deposit \( \$ 10000 \) into the account. 

This can be modeled with the differential equation: 

\( \frac { d y } { d t } = 0.05 y + 10000 \) 

\( y ( 0 ) = 50000 \) 

Solve this differential equation for \( y ( t ) \) 

\( y ( t ) = \) 

Let \( y ( t ) \) represent your bank account balance, in dollars, after \( t \) years. Suppose you start with \( \$ 50000 \) in the account. Each year the account earns \( 5 \% \) interest, and you deposit \( \$ 10000 \) into the account. 

This can be modeled with the differential equation: 

\( \frac { d y } { d t } = 0.05 y + 10000 \) 

\( y ( 0 ) = 50000 \) 

Solve this differential equation for \( y ( t ) \) . 

An inverted pyramid is being filled with water at a constant rate of \( 65 \) cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length \( 3 cm \) , and the height is \( 6 cm \) . 

Find the rate at which the water level is rising when the water level is \( 4 cm \) . 

Evaluate the integral below by interpreting it in terms of areas in the figure. 

The areas of the labeled regions are 

\( A 1 = 7 , A 2 = 3 , A 3 = 1 \) and \( A 4 = 1\) 

\(V = \int _ { 0 } ^ { 7 } f ( x ) d x\)

For the given parametric function find derivatives of the first and second orders 

\( x = a \cos t , y = b \sin t\) 

Which answer best describes the following differential equation \(\frac { d ^ { 4 } y } { d x ^ { 4 } } + \frac { d y } { d x } = y ^ { 6 } \sin x ?\) 

(A) \(6\) th order, linear, homogeneous 

(B) 4th order, nonlinear, homogeneous 

(C) 6th order, nonlinear, homogeneous 

(D) 4th order, nonlinear, inhomogeneous 

(E) \(6\) th order, nonlinear, homogeneous 

(F) \(6\) th order, linear, inhomogeneous 

(G) 4th order, linear, inhomogeneous linear, homogeneous 

(H) \(6\) th order, linear, inhomogeneous 

The total revenue from sales of ski pants is modeled by the following expression:\( 720 + 4 x - 2 x ^ { 2 } \) . Revenue is also calculated using the number of pants sold and the price per pant. Determine expressions for the number of pants sold and the price per pants. 

Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.) 

\(\lim\limits_{x\to3}\frac { x ^ { 3 } - 27 } { x - 3 } \)

Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result. 

\( g ( x ) = \) _____

Find \( \frac { d y } { d x } \) if  \(y = \frac { 6 x ^ { 4 } + 8 x ^ { 3 } } { 2 x ^ { 2 } }\) Remember: split up the numerator using the property \( \frac { A + B } { C } = \frac { A } { C } + \frac { B } { C } \) , then reduce each fraction and take the Power Rule.

A. \(18 x ^ { 5 } + 4 x \)  

B. \( 8 x + 2 \) 

C. \( 24 x ^ { 3 } + 4\) 

D. \(6x+ 4\)

This question is designed to be answered with a calculator. An object's velocity is given by \( v ( t ) = 0.5 ( t + 3 ) ( t - 1 ) ( t - 5 ) ^ { 2 } \) , where \( t \) is time, in seconds. What is the total distance traveled by the object in the first \( 5 \) minutes? 

\( 37.500 \) 

\( 41.667 \) 

\( 59.733 \) 

\( 77.800\) 

This question is designed to be answered with a calculator. An object's velocity is given by \( v ( t ) = 0.5 ( t + 3 ) ( t - 1 ) ( t - 5 ) ^ { 2 } \) , where \( t \) is time, in seconds. What is the total distance traveled by the object in the first \( 5 \) minutes? 

\( 37.500 \) 

\( 41.667 \) 

\( 59.733 \) 

\( 77.800\) 

This question is designed to be answered with a calculator.

 The population, \( P \) , of a city is changing at a rate \( \frac { d P } { d t } = 0.012 P \) , in people per year. Approximately how many years will it take for the population to double? 

\( 57.762 \) 

- \( 58.108 \) 

\( 83.333 \) 

\( 166.667\) 

Which of the following statement is false for a vertical reflection?

 a. The invariant points are the \( x \) -intercepts of the original function. 

b. A vertical reflection on \( f ( x ) \) becomes the same as \( f ( - x ) \) . 

c. All invariant points will have a \( y \) -value of \( 0 \) . 

d. A vertical reflection on \( f ( x ) \) becomes the same as \( - f ( x ) \) . 

Find the critical numbers, the intervals on which \( f ( x ) \) is increasing, the intervals on which \( f ( x ) \) is decreasing, and the local extrema. Do not graph. 

\( f ( x ) = \frac { x ^ { 2 } } { x - 2 } \) 

Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The critical number(s) of the function are \( x = \) (Use a comma to separate answers as needed.) B. There is no solution. 

Use the position function \( s ( t ) = - 4.9 t ^ { 2 } + 750 , \) which gives the height (in meters) of an object that has fallen for following. 

\( \lim _ { t \rightarrow a } \frac { s ( a ) - s ( t ) } { a - t }\) 

Find the velocity of the object when \( t = 2 \) . 

Given the function \( f ( x ) = x ^ { 2 } - 14 , x \geq 0 \) , complete parts a through c. (

a) Find an equation for \( f ^ { - 1 } ( x ) \) . 

(a) Find \( f ^ { - 1 } ( x ) \) . 

\( f ^ { - 1 } ( x ) = \square \) 

(Type an exact answer. using radicals as needed.) 

Find the GCF of each pair of monomials. 

8. \( 12 y ^ { 3 } \) and \( 15 y ^ { 2 } \) 

9. \( 3 p ^ { 4 } \) and \( 4 p \) 

10. \( 18 x ^ { 6 } \) and \( 24 y ^ { 2 } \) 

11. \( 14 x y ^ { 2 } \) and \( 21 y ^ { 3 }\) 

Assume that \( x \) and \( y \) are both differentiable functions of \( t \) and find the required values of \( d y / d t \) and \( d x / d t \) . 

\( x ^ { 2 } + y ^ { 2 } = 400\) 

(a) Find \( d y / d t \) , given \( x = 12 , y = 16 , \) and \( d x / d t = 5 \) 

\( d y / d t = \) 

(b) Find \( d x / d t , \) given \( x = 16 , y = 12 , \) and \( d y / d t = - 2 \) 

\( d x / d t = \) 

Find \( d y / d x \) implicitly and find the largest interval of the form \( - a < y < a \) or \( 0 < y < a \) such that \( y \) is a differentiable function of x

\( \cos y = x\) 

\(\square < y < \) \(\square\)

Write \( d y / d x \) as a function of \( x \) . 

\(\frac { d y } { d x } = \square\)

If the distance a runner travels is represented by \( d ( h ) = 2 \sqrt { h } \) and the associated time is \( t ( h ) = 3 \sqrt { 4 h } \) , what is the runner's speed? 

\(s ( h ) = \frac { \sqrt { 2 } } { h } \) 

\( s ( h ) = 12 h \) 

\( s ( h ) = h \sqrt { 3 } \) 

\( s ( h ) = \frac { 1 } { 3 }\) 

Determine the end behavior of the radical function. 

\( y = 2 \sqrt { x + 2 } + 4 \) 

As \( x \rightarrow \infty , f ( x ) \rightarrow \infty \) . As \( x \rightarrow - \infty , f ( x ) \rightarrow - \infty \) 

As \( x \rightarrow \infty , f ( x ) \rightarrow - \infty \) . As \( x \rightarrow - 2 , f ( x ) \rightarrow 4 \) 

As \( x \rightarrow \infty , f ( x ) \rightarrow \infty \) . As \( x \rightarrow - 2 , f ( x ) \rightarrow 4 \) 

As \( x \rightarrow \infty , f ( x ) \rightarrow \infty \) . As \( x \rightarrow - 2 , f ( x ) \rightarrow - \infty\) 

Drag the tiles to the correct boxes to complete the pairs. Match the one-to-one functions with their inverse functions. 

\(f ( x ) = \frac { 2 x } { 3 } - 17 \quad f ( x ) = x - 10 \quad f ( x ) = \sqrt[ 3 ] { 2 x } \quad f ( x ) = \frac { x } { 5 }\)

Inverse Function                                      Function 

\(f ^ { - 1 } ( x ) = 5 x\)----------------------> A.

\(f ^ { - 1 } ( x ) = \frac { x ^ { 3 } } { 2 }\)----------------------> B.

\(f ^ { - 1 } ( x ) = x + 10\)--------------> C. 

\(f ^ { - 1 } ( x ) = \frac { 3 ( x + 17 ) } { 2 }\)--------------> D.

As researchers study the HPV virus in the laboratory, use the approximate doubling time formula to calculate the time needed for \( 250 \) virus germs to double if the growing rate is \( 5 \% \) per hour. Choose the correct answer below.

A. \( 12.5 \) hours 

B. \( 50 \) hours

C. \( 14 \) hours 

D. \( 2.8 \) hours 

Express your answer as a decimal rounded to three decimal places, if necessary. Find the average value of the function on the given interval. 

\( f ( x ) = x ^ { 2 } - 2 , [ 2,4 ] \) 

\( f _ { a v e } = \) 

Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) 

\( f ( x ) = x \sqrt { x + 6 } \) 

\( ( x , y ) = ( \square)\) 

Describe the concavity. (Enter your answers using interval notation.) 

concave upward 

concave downward 

Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) 

\( f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 14 x + 1 \) 

\(( x , y ) = (\square)\)

Discuss the concavity of the graph of the function. (Enter your answers using interval notation.) concave upward 

concave downward 

\(\frac { d y } { d x } = x + 1 , y ( 2 ) = 3 \) 

\( y = \frac { x ^ { 2 } } { 2 } + x + 1 \) 

\( y = \frac { x ^ { 2 } } { 2 } + x - 2 \) 

\( y = \frac { x ^ { 2 } } { 2 } + x - 1 \) 

\( y = \frac { x ^ { 2 } } { 2 } + x + 2\) 

Remember: there is no " \( + C ^ { \prime \prime } \) here. We integrate, then evaluate between the upper and lower limit, subtracting the lower limit from the upper limit. Choose one \.

 \(\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } 2 \cos ( x ) d x\)

Write your answer as a reduced improper fraction if needed. Evaluate the following integral: 

\( \int _ { 0 } ^ { 1 } \frac { 6 x } { ( x ^ { 2 } + 1 ) ^ { 2 } } d x = \) 

Find the indefinite integral as indicated. 

\( \int \frac { d } { d x } ( x ^ { 5 } + 3 x ^ { 2 } + 8 ) d x \) 

\( \int \frac { d } { d x } ( x ^ { 5 } + 3 x ^ { 2 } + 8 ) d x = \) 

Anna is buying a 255,000 dollar house. she has. to give a 20 % down payment . sheet got a. 30 yr mortage with a interest rate of 5 %. Determine the monthly. payment for principal and interest? her monthly payment for principal and interest is ?

The volume of a crate with length of \( x \) feet, width of \( 8 - x \) feet, and a height of \( x - 3 \) feet is given by \( V ( x ) = x ( 8 - x ) ( x - 3 ) \) . Find the maximum volume of the crate and the dimensions for this volume. 

Find \( \frac { d y } { d x } \) 

\( y = \int _ { \sqrt { x } } ^ { \pi } \sin ( t ^ { 2 } ) d t \) 

\( \frac { d y } { d x } = \square\) 

Consider the graph of the function \( f ( x ) = - x ^ { 2 } + 8 \) (see figure). (a) Find the equation of the secant line joining the points \( ( - 1,7 ) \) and \( ( 2,4 ) \) . (b) Use the Mean Value Theorem to determine a point \( c \) in the interval \( ( - 1,2 ) \) such that the tangent line at \( c \) is parallel to the secant line. (c) Find the equation of the tangent line through \( c \) . 

Which statement could be used to explain why \( f ( x ) = 2 x - 3 \) has an inverse relation that is a function? 

The graph of \( f ( x ) \) passes the vertical line test. 

\( f ( x ) \) is a one-to-one function. 

The graph of the inverse of \( f ( x ) \) passes the horizontal line test. 

\( f ( x ) \) is not a function. 

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the \( x \) -axis over the given interval. 

\( g ( x ) = 2 x ^ { 2 } - x - 1 , [ 2,4 ] , 4 \) rectangles Need Help? Area \( < \) Read It 

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the \( x \) -axis over the given interval. (Round your answers to four decimal places.) 

\( g ( x ) = 3 \sin x , [ 0 , \pi ] , 6 \) rectangles left endpoints approximation Area: right endpoints approximation Area: 

Use implicit differentiation to find \( y ^ { \prime } \) , and then evaluate \( y ^ { \prime } \) for \( x ^ { 2 } y - 4 x ^ { 2 } = 0 \) at the point \( ( 2,4 ) \) . 

\( y ^ { \prime } = \square \) 

\( y ^ { \prime } | _ { ( 2,4 ) } = \square \) (Simplify your answer.) 

Given the function \( f ( x ) = 4 x - 5 , \) evaluate and simplify the expressions below. See special instructions on how to enter your answers. 

\( f ( a ) = \square\)  ,\(f ( a + h ) = \square\)

\( \frac { f ( a + h ) - f ( a ) } { h } = \square\) Instructions: Simplify answers as much as possible. Expressions such as \( 4 ( x + 2 ) \) and \( ( x + 5 ) ^ { 2 } \) should be expanded. Also collect like terms, so \( 3 x + x \) should be written as \( 4 x \) . 

 The function \( f ( x ) = \frac { 5 x + 4 } { x - 9 } \) is one-to-one. 

a. Find an equation for \( f ^ { - 1 } ( x ) \) , the inverse function.

 b. Verify that your equation is correct by showing that \( f ( f ^ { - 1 } ( x ) ) = x \) and \( f ^ { - 1 } ( f ( x ) ) = x \) . a. Select the correct choice below and fill in the answer box \( ( \) es \( ) \) to complete your choice. (Simplify your answers. Use integers or fractions for any numbers in the expression.) A. \( f ^ { - 1 } ( x ) = \square \) , for \( x \neq \square \) 

B. \( f ^ { - 1 } ( x ) = \square \) , for \( x \leq \square \) , for all \( x \) c. \( f ^ { - 1 } ( x ) = \square \) , for \( x \geq \square\) 

Find the intervals on which \( f ( x ) \) is increasing, the intervals on which \( f ( x ) \) is decreasing, and the local extrema. 

Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Type an integer or decimal rounded to four decimal places as needed. Use a comma to separate answers as needed.) A. The function has local maximum \( f ( c ) = \) and no local minimum. B. The function has local minimum \( f ( c ) = \) and no local maximum. C. The function has local maximum \( f ( c ) = \) and local minimum \( f ( c ) = \) 

D. The function has no local extrema. 

If an object moves along the \( y \) -axis (marked in feet) so that its position at time \( x \) (in seconds) is given by \( f ( x ) = 182 x - 13 x ^ { 2 } \) , find the following. 

(A) The instantaneous velocity function \( v = f ^ { \prime } ( x ) \) 

(B) The velocity when \( x = 0 \) and \( x = 4 \) sec (C) The time(s) when \( v = 0 \) 

(A) \( v ( x ) = 182 - 26 x \) 

(B) The velocity when \( x = 0 \sec \) is \( \square ft / sec \) . The velocity when \( x = 4 \sec \) is \( \square ft / sec \) . 

Find \( F \) as a function of \( x \) and evaluate it at \( x = 3 , x = 7 , \) and \( x = 9 \) 

\( F ( x ) = \int _ { 3 } ^ { x } - \frac { 2 } { t ^ { 3 } } d t \) 

\( F ( 3 ) = \) 

\( F ( 7 ) = \) 

\( F ( 9 ) = \) 

The graph of \( f \) consists of line segments, as shown in the figure. Evaluate each definite integral by using geometric formulas. 

(a) \( \int _ { 0 } ^ { 1 } - 6 f ( x ) d x\) 

(b) \( \int _ { 3 } ^ { 4 } 8 f ( x ) d x\) 

(c) \( \int _ { 0 } ^ { 7 } 4 f ( x ) d x \) 

(d) \( \int _ { 5 } ^ { 11 } - 7 f ( x ) d x \) 

(e) \( \int _ { 0 } ^ { 11 } 2 f ( x ) d x \) 

(f) \( \int _ { 4 } ^ { 10 } 3 f ( x ) d x\) 

A projectile is fired vertically, and its height (in feet) after \( t \) seconds is given by \( s ( t ) = - 3 t ^ { 2 } + 12 t + 756 . \) 

Step \( 1 \) of \( 3 \) : Find the maximum height of the projectile. Round your final answer to the nearest hundredth.

Calculate the definite integral by referring to the figure with the indicated areas. Area \( A = 1.367 \) Area \( C = 5.813 \) 

Area \( B = 2.306 \) Area \( D = 1.849\) 

Which of the following is true? 

a) \( \Gamma ( n + 1 ) = n \Gamma ( n ) \) for any real number 

b) \( \Gamma ( n ) = n \Gamma ( n + 1 ) \) for any real number 

c) \( \Gamma ( n + 1 ) = n \Gamma ( n ) \) for \( n > 1 \) 

d) \( \Gamma ( n ) = n \Gamma ( n + 1 ) \) for \( n > 1\) 

If  \(x ^ { 4 } \cdot y - x \cdot y = 14\) 

What is the value of the \(\frac { d y } { d x }\) at the point \(( 2,1 ) ?\) 

4. Two buses use two different routes to travel from station \( A \) to station \( B \) . First bus reaches at \( B \) via point \( C \) and second bus reaches at \( B \) via point \( D \) . If coordinates of \( A , B , C \) and \( D \) are \( ( - 2,3 ) , ( 2,3 ) , ( 3,2 ) \) and \( ( - 1,0 ) \) respectively, then which bus will arrive at station \( B \) first? (assuming both buses have same speed). (5 marks) 

Use L'Hôpital's Rule (possibly more than once) to evaluate the following limit 

\( \lim _ { t \rightarrow 0 } ( 4 \sin ( 11 t ) \ln ( 11 t ) ) = \) 

If the answer equals \( \infty \) or \( - \infty \) , write INF or -INF in the blank. 

Find the area under the curve \( y = f ( x ) \) for \( x \in [ 0,7 \pi ]\) 

Hint: You shouldn't need to calculate all of these integrals. Sketch the curve and see if you can relate to certain areas. 

 \( f ( x ) = \left\{ \begin{array} { l } { \sin ( x ) , x \in [ 0 , \pi ] } \\ { \frac { x } { \pi ^ { 2 } } , x \in [ \pi , 2 \pi ] } \\ { \sin ( x ) \in [ 2 \pi , 3 \pi ] } \\ { \frac { 2 x ^ { 2 } } { \pi ^ { 3 } } , x \in [ 3 \pi , 4 \pi ] } \\ { \sin ( x ) , x \in [ 4 \pi , 5 \pi ] } \\ { 0 , x \in [ 5 \pi , 6 \pi ] } \\ { \sin ( x ) , x \in [ 6 \pi , 7 \pi ] } \end{array} \right.\) 

Find the area under one hump of the sine curve \( f ( x ) = 6 \cdot \sin ( x ) \) . In other words, find the area under the graph of the function \( f ( x ) = 6 \cdot \sin ( x ) \) bounded by the \( x \) -axis and the lines \( x = 0 \) and \( x = \pi \) . 

Area: _____

Q-2: [6+4 marks] 

a) Let \( E = \left[ \begin{array} { l l l } { 1 } & { 2 } & { 1 } \\ { 0 } & { 1 } & { 3 } \\ { 1 } & { 0 } & { r } \end{array} \right] \) . Find all values of \( r \) for which \( E \) has inverse. Find, if possible, \( E ^ { - 1 } \) for \( r = - 1 \) . 

b) Let \( A = \left[ \begin{array} { l l } { 0 } & { 2 } \\ { 0 } & { 3 } \end{array} \right] \) . Find all possible \( 2 \times 2 \) matrices \( B \) such that \( A B = O _ { 2 }\) 

Question 4: (5 points) 

A sector of a circle has a diameter of \( 20 \) feet and an angle of \( \frac { 2 \pi } { 3 } \) radians. Find the area of the sector. Round your answer to four decimal places. 

\( A = \) ______ft\(^{2} \)

\(\int _ { 6 } ^ { 10 } ( f ( x ) + g ( x ) ) d x = 10 , \int _ { 10 } ^ { 6 } g ( x ) d x = 3\) 

and \(\int _ { 10 } ^ { 0 } 3 f ( x ) d x = - 45\) 

find \(\int _ { 0 } ^ { 6 } f ( x ) d x\) 

I have discovered the secret recipe for Froggatt's Lumpy Sprout Ketchup, which needs \( 342 \) mouldy sprouts to make \( 1 \) pot. I have decided to make \( 6 \) pots, to last me through the long cold winter. How many mouldy sprouts will I need? 

if \(\int _ { 6 } ^ { 10 } ( f ( x ) + g ( x ) ) d x = 10 , \int _ { 10 } ^ { 6 } g ( x ) d x = 3 \) 

and \( \int _ { 10 } ^ { 0 } 3 f ( x ) d x = - 45 \),

find \( \int _ { 0 } ^ { 6 } f ( x ) d x\) 

Let \( F ( x ) = f ( f ( x ) ) \) and \( G ( x ) = ( F ( x ) ) ^ { 2 } \) 

You also know that \( f ( 4 ) = 8 , f ( 8 ) = 3 , f ^ { \prime } ( 8 ) = 10 , f ^ { \prime } ( 4 ) = 12 \) 

Find \( F ^ { \prime } ( 4 ) = \square \) and \( G ^ { \prime } ( 4 ) = \square\) 

6. Naomi put \( \$ 4600 \) in the bank to save for a sports car. If she was paid \( 7 \% \) interest on this amount only each year, how much interest did she earn after: 

a. \( 1 \) year 

\(\left. \begin{array} { l } { \text { Some computer algebra systems have commands that will } } \\ { \text { draw approximating rectangles and evaluate the sums of } } \\ { \text { their areas, at least if } x _ { i } ^ { * } \text { is a left or right endpoint. (For } } \\ { \text { instance, in Maple use leftbox, rightbox, left- } } \\ { \text { sum, and rightsum.) } } \\ { \text { (a) If } f ( x ) = 1 / ( x ^ { 2 } + 1 ) , 0 \leq x \leq 1 \text { , find the left and } } \\ { \text { right sums for } n = 10,30 \text { , and } 50 \text { . } } \\ { \text { (b) Illustrate by graphing the rectangles in part (a). } } \\ { \text { (c) Show that the exact area under } f \text { lies between } 0.780 } \\ { \text { and } 0.791 \text { . } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { Evaluate } \int \int \int _ { E } z d V \text { , where } E \text { lies above the paraboloid } } \\ { z = x ^ { 2 } + y ^ { 2 } \text { and below the plane } z = 2 y \text { . Use either the } } \\ { \text { Table of Integrals (on Reference Pages } 6 - 10 \text { ) or a computer } } \\ { \text { algebra system to evaluate the integral. } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { The pressure of } 1 \text { mole of an ideal gas is increasing at a rate } } \\ { \text { of } 0.05 kPa / s \text { and the temperature is increasing at a rate of } } \\ { 0.15 K / s \text { . Use the equation } P V = 8.31 T \text { in Example } 2 \text { to find } } \\ { \text { the rate of change of the volume when the pressure is } 20 kPa } \\ { \text { and the temperature is } 320 K \text { . } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { A force of } 10 lb \text { is required to hold a spring stretched } 4 \text { in. } } \\ { \text { beyond its natural length. How much work is done in stretch- } } \\ { \text { ing it from its natural length to } 6 \text { in. beyond its natural length? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { The von Bertalanffy growth model is used to predict the length } } \\ { L ( t ) \text { of a fish over a period of time. If } L _ { \infty } \text { is the largest length } } \\ { \text { for a species, then the hypothesis is that the rate of growth in } } \\ { \text { length is proportional to } L _ { \infty } - L \text { , the length yet to be achieved. } } \\ { \text { (a) Formulate and solve a differential equation to find an } } \\ { \text { expression for } L ( t ) \text { . } } \\ { \text { (b) For the North Sea haddock it has been determined that } } \\ { L _ { \infty } = 53 cm , L ( 0 ) = 10 cm \text { , and the constant of propor- } } \\ { \text { tionality is } 0.2 \text { . What does the expression for } L ( t ) \text { become } } \\ { \text { with these data? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { The body mass index (BMI) of a person is defined by } } \\ { \qquad B ( m , h ) = \frac { m } { h ^ { 2 } } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { A retailer has been selling } 1200 \text { tablet computers a week } } \\ { \text { at } \$ 350 \text { each. The marketing department estimates that an } } \\ { \text { additional } 80 \text { tablets will sell each week for every } \$ 10 \text { that } } \\ { \text { the price is lowered. } } \\ { \text { (a) Find the demand function. } } \\ { \text { (b) What should the price be set at in order to maximize } } \\ { \text { revenue? } } \\ { \text { (c) If the retailer's weekly cost function is } } \\ { \qquad C ( x ) = 35,000 + 120 x } \\ { \text { what price should it choose in order to maximize its } } \\ { \text { profit? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { (a) Estimate the volume of the solid that lies below the surface } } \\ { z = x y \text { and above the rectangle } } \\ { \qquad R = \{ ( x , y ) | 0 \leq x \leq 6,0 \leq y \leq 4 \} } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { If a bacteria population starts with } 100 \text { bacteria and doubles } } \\ { \text { every three hours, then the number of bacteria after } t \text { hours } } \\ { \text { is } n = f ( t ) = 100 \cdot 2 ^ { t / 3 } \text { . } } \\ { \text { (a) Find the inverse of this function and explain its meaning. } } \\ { \text { (b) When will the population reach } 50,000 \text { ? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { (a) Use the Midpoint Rule for double integrals (see Sec- } } \\ { \text { tion } 15.1 \text { ) with four squares to estimate the surface area } } \\ { \text { of the portion of the paraboloid } z = x ^ { 2 } + y ^ { 2 } \text { that lies } } \\ { \text { above the square } [ 0,1 ] \times [ 0,1 ] \text { . } } \\ { \text { (b) Use a computer algebra system to approximate the sur- } } \\ { \text { face area in part (a) to four decimal places. Compare } } \\ { \text { with the answer to part (a). } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { (a) A company makes computer chips from square wafers } } \\ { \text { of silicon. It wants to keep the side length of a wafer } } \\ { \text { very close to } 15 mm \text { and it wants to know how the area } } \\ { A ( x ) \text { of a wafer changes when the side length } x \text { changes. } } \\ { \text { Find } A ^ { \prime } ( 15 ) \text { and explain its meaning in this situation. } } \\ { \text { (b) Show that the rate of change of the area of a square } } \\ { \text { with respect to its side length is half its perimeter. Try } } \\ { \text { to explain geometrically why this is true by drawing a } } \\ { \text { square whose side length } x \text { is increased by an amount } } \\ { \Delta x . \text { How can you approximate the resulting change in } } \\ { \text { area } \Delta A \text { if } \Delta x \text { is small? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { The speeds of vehicles on a highway with speed limit } } \\ { 100 km / h \text { are normally distributed with mean } 112 km / h \text { and } } \\ { \text { standard deviation } 8 km / h \text { . } } \\ { \text { (a) What is the probability that a randomly chosen vehicle is } } \\ { \text { traveling at a legal speed? } } \\ { \text { (b) If police are instructed to ticket motorists driving } } \\ { 125 km / h \text { or more, what percentage of motorists are } } \\ { \text { targeted? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { Suppose that the graph of } y = \log _ { 2 } x \text { is drawn on a coordi- } } \\ { \text { nate grid where the unit of measurement is an inch. How } } \\ { \text { many miles to the right of the origin do we have to move } } \\ { \text { before the height of the curve reaches } 3 ft \text { ? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { The kinetic energy KE of an object of mass } m \text { moving with } } \\ { \text { velocity } v \text { is defined as } KE = \frac { 1 } { 2 } m v ^ { 2 } \text { . If a force } f ( x ) \text { acts } } \\ { \text { on the object, moving it along the } x \text { -axis from } x _ { 1 } \text { to } x _ { 2 } \text { , the } } \\ { \text { Work-Energy Theorem states that the net work done is equal } } \\ { \text { to the change in kinetic energy: } \frac { 1 } { 2 } m v _ { 2 } ^ { 2 } - \frac { 1 } { 2 } m v _ { 1 } ^ { 2 } \text { , where } v _ { 1 } \text { is } } \\ { \text { the velocity at } x _ { 1 } \text { and } v _ { 2 } \text { is the velocity at } x _ { 2 } \text { . } } \\ { \text { (a) Let } x = s ( t ) \text { be the position function of the object at } } \\ { \text { time } t \text { and } v ( t ) , a ( t ) \text { the velocity and acceleration func- } } \\ { \text { tions. Prove the Work-Energy Theorem by first using } } \\ { \text { the Substitution Rule for Definite Integrals (4.5.5) to } } \\ { \text { show that } } \\ { \qquad W = \int _ { x _ { 1 } } ^ { x _ { 2 } } f ( x ) d x = \int _ { t _ { 1 } } ^ { t _ { 2 } } f ( s ( t ) ) v ( t ) d t } \\ { \text { Then use Newton's Second Law of Motion } } \\ { \text { (force } = \text { mass } \times \text { acceleration) and the substitution } } \\ { \qquad u = v ( t ) \text { to evaluate the integral. } } \\ { \text { (b) How much work (in ft-lb) is required to hurl a } 12 \text { -lb } } \\ { \text { bowling ball at } 20 \text { mi/h? (Note: Divide the weight } } \\ { \text { in pounds by } 32 ft / s ^ { 2 } \text { , the acceleration due to gravity, } } \\ { \text { to find the mass, measured in slugs.) } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { In Exercise } 14.1 .39 \text { and Example } 14.3 .3 \text { , the body mass index } } \\ { \text { of a person was defined as } B ( m , h ) = m / h ^ { 2 } \text { , where } m \text { is the } } \\ { \text { mass in kilograms and } h \text { is the height in meters. } } \\ { \text { (a) What is the linear approximation of } B ( m , h ) \text { for a child } } \\ { \text { with mass } 23 kg \text { and height } 1.10 m \text { ? } } \\ { \text { (b) If the child's mass increases by } 1 kg \text { and height by } 3 cm \text { , } } \\ { \text { use the linear approximation to estimate the new BMI. } } \\ { \text { Compare with the actual new BMI. } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { A Cepheid variable star is a star whose brightness alternately } } \\ { \text { increases and decreases. The most easily visible such star is } } \\ { \text { Delta Cephei, for which the interval between times of max- } } \\ { \text { imum brightness is } 5.4 \text { days. The average brightness of this } } \\ { \text { star is } 4.0 \text { and its brightness changes by } \pm 0.35 \text { . In view of } } \\ { \text { these data, the brightness of Delta Cephei at time } t \text { , where } } \\ { t \text { is measured in days, has been modeled by the function } } \\ { \qquad B ( t ) = 4.0 + 0.35 \sin ( \frac { 2 \pi t } { 5.4 } ) } \\ { \text { (a) Find the rate of change of the brightness after } t \text { days. } } \\ { \text { (b) Find, correct to two decimal places, the rate of increase } } \\ { \text { after one day. } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { (a) Find the volume of the region } E \text { that lies between } } \\ { \text { the paraboloid } z = 24 - x ^ { 2 } - y ^ { 2 } \text { and the cone } } \\ { z = 2 \sqrt { x ^ { 2 } + y ^ { 2 } } \text { . } } \\ { \text { (b) Find the centroid of } E \text { (the center of mass in the case } } \\ { \text { where the density is constant). } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { A canister is dropped from a helicopter } 500 m \text { above the } } \\ { \text { ground. Its parachute does not open, but the canister has } } \\ { \text { been designed to withstand an impact velocity of } 100 m / s \text { . } } \\ { \text { Will it burst? } } \end{array} \right.\)

\(\left. \begin{array} { l } { \text { Scientists can determine the age of ancient objects by the } } \\ { \text { method of radiocarbon dating. The bombardment of the upper } } \\ { \text { atmosphere by cosmic rays converts nitrogen to a radioactive } } \\ { \text { isotope of carbon, } { } ^ { 14 } C \text { , with a half-life of about } 5730 \text { years. } } \\ { \text { Vegetation absorbs carbon dioxide through the atmosphere and } } \\ { \text { animal life assimilates } { } ^ { 14 } C \text { through food chains. When a plant } } \\ { \text { or animal dies, it stops replacing its carbon and the amount of } } \\ { { } ^ { 14 } C \text { begins to decrease through radioactive decay. Therefore } } \\ { \text { the level of radioactivity must also decay exponentially. } } \\ { \text { A discovery revealed a parchment fragment that had about } } \\ { 74 \% \text { as much } { } ^ { 14 } C \text { radioactivity as does plant material on the } } \\ { \text { earth today. Estimate the age of the parchment. } } \end{array} \right.\)