A function \( f ( x ) \) and interval \( [ a , b ] \) are given. Check if the Mean Value Theorem can be applied to \( f \) on \( [ a , b ] \) . If so, find all values \( c \) in \( [ a , b ] \) guaranteed by the Mean value Theorem Note, if the Mean value Theorem does not apply, enter DNE for the \( c \) value.
\(f( x ) = \sin ^ { - 1 } x \) on \( [ - 1,1 ]\)
\(\sqrt{1- \frac{4}{\pi ^2} }\), \(- \sqrt{1- \frac{4}{\pi ^2} }\)
Find the solution of the exponential equation \( 16 e ^ { x } = 12 \) in terms of logarithms, or correct to four decimal places. \( x = \)
Certain radioactive material decays in such a way that the mass remaining after \( t \) years is given by the function
\( m ( t ) = 100 e ^ { - 0.01 t } \)
where \( m ( t ) \) is measured in grams. (a) Find the mass at time \( t = 0 \) . Your answer is (b) How much of the mass remains after \( 20 \) years? Your answer is Round answers to \( 1 \) decimal place.
![A function \( f ( x ) \) and interval \( [ a , b ]...](https://static.cameramath.com/jkyx9y2yhe/d94ac47b8e35a01d9f27d9cc4e65f7b0)
