Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 54CR
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do number 9 pls and pls show all the work
![### Calculus Practice Problems
1. **Find the antiderivative of \( f'(x) \). Initial condition is \( f(0) = 3 \). Let \( f'(x) = 4x^3 + x^2 - 3x \).**
2. **Find the function \( g(x) \) where \( g'(x) = 4 \sin x + 7x - \frac{5}{x} \)**
3. **Find \( y' \) and then evaluate it for \((-2, 3)\)**
\[
3x^3 + y^2 = -xy
\]
4. **Find the limit:**
\[
\lim_{{x \to 0}} \frac{e^{2x} - 1}{{\cos x - 1}}
\]
5. **Find absolute maximum and absolute minimum of**
\[
f(x) = x^3 - 3x^2 + 1 \text{ with the domain of } [-2, 4]
\]
6. **Compute \( \nabla y \) and \( \nabla y \) for \( f(x) = x - x^3 \) where \( x = 1 \) and \( \nabla x = 0.1 \)**
7. **Air is being pumped into a spherical balloon so that the volume increases at a rate of 50 cubic feet per second. How fast is the radius of the balloon increased when the diameter is 26 feet?**
**Given:**
- Diameter = 2 times the radius.
- Circle Area = \( \pi r^2 \).
**Formula:**
\[
\text{Sphere Volume:} \quad v = \frac{4}{3}\pi r^3
\]
8. **Find the limit. Must use l'Hospital's Rule.**
\[
\lim_{{x \to \infty}} \frac{\ln x}{x - 1}
\]
9. **Find the equation of the tangent line to \( y = 2t \sin t \) at point \( \left(\frac{\pi}{2}, \pi \right) \)**
10. **Find the derivative:**
\[
\frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b3472e-ab59-40f2-b296-ac3727a4d9f3%2F19b6558b-f52c-477d-852f-f8fe04de82de%2Fok1r1lqs_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Practice Problems
1. **Find the antiderivative of \( f'(x) \). Initial condition is \( f(0) = 3 \). Let \( f'(x) = 4x^3 + x^2 - 3x \).**
2. **Find the function \( g(x) \) where \( g'(x) = 4 \sin x + 7x - \frac{5}{x} \)**
3. **Find \( y' \) and then evaluate it for \((-2, 3)\)**
\[
3x^3 + y^2 = -xy
\]
4. **Find the limit:**
\[
\lim_{{x \to 0}} \frac{e^{2x} - 1}{{\cos x - 1}}
\]
5. **Find absolute maximum and absolute minimum of**
\[
f(x) = x^3 - 3x^2 + 1 \text{ with the domain of } [-2, 4]
\]
6. **Compute \( \nabla y \) and \( \nabla y \) for \( f(x) = x - x^3 \) where \( x = 1 \) and \( \nabla x = 0.1 \)**
7. **Air is being pumped into a spherical balloon so that the volume increases at a rate of 50 cubic feet per second. How fast is the radius of the balloon increased when the diameter is 26 feet?**
**Given:**
- Diameter = 2 times the radius.
- Circle Area = \( \pi r^2 \).
**Formula:**
\[
\text{Sphere Volume:} \quad v = \frac{4}{3}\pi r^3
\]
8. **Find the limit. Must use l'Hospital's Rule.**
\[
\lim_{{x \to \infty}} \frac{\ln x}{x - 1}
\]
9. **Find the equation of the tangent line to \( y = 2t \sin t \) at point \( \left(\frac{\pi}{2}, \pi \right) \)**
10. **Find the derivative:**
\[
\frac
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