Garrett Nunez
05/23/2024 · Junior High School
Calule la razón de cambio instantánea \( \lim _{x \rightarrow x_{0}} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}} \), si \( f(x)=\cos (x) \), y \( x_{0}=\frac{4 \pi}{3} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Calculate the limit \( \lim _{x \rightarrow \frac{4 \pi}{3}} \frac{\cos(x)-\cos\left(\frac{4 \pi}{3}\right)}{x-\frac{4 \pi}{3}} \).
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
\(\lim _{x\rightarrow \frac{4\pi }{3}}\left(\frac{\cos\left(x\right)-\cos\left(\frac{4\pi }{3}\right)}{x-\frac{4\pi }{3}}\right)\)
- step1: Use the L'Hopital's rule:
\(\lim _{x\rightarrow \frac{4\pi }{3}}\left(\frac{\frac{d}{dx}\left(\cos\left(x\right)-\cos\left(\frac{4\pi }{3}\right)\right)}{\frac{d}{dx}\left(x-\frac{4\pi }{3}\right)}\right)\)
- step2: Find the derivative:
\(\lim _{x\rightarrow \frac{4\pi }{3}}\left(\frac{-\sin\left(x\right)}{1}\right)\)
- step3: Simplify the expression:
\(\lim _{x\rightarrow \frac{4\pi }{3}}\left(-\sin\left(x\right)\right)\)
- step4: Rewrite the expression:
\(-\lim _{x\rightarrow \frac{4\pi }{3}}\left(\sin\left(x\right)\right)\)
- step5: Calculate:
\(-\left(-\frac{\sqrt{3}}{2}\right)\)
- step6: Use the rules for multiplication and division:
\(\frac{\sqrt{3}}{2}\)
La razón de cambio instantánea \( \lim _{x \rightarrow \frac{4 \pi}{3}} \frac{\cos(x)-\cos\left(\frac{4 \pi}{3}\right)}{x-\frac{4 \pi}{3}} \) es \( \frac{\sqrt{3}}{2} \).
Quick Answer
La razón de cambio instantánea es \( \frac{\sqrt{3}}{2} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit