(1) Find the antiderivative of f(x). Initial condition is f(0) =3. Let f'(x) = 4x³ + x² – 3x (2) Find the function g(x) where g'(x) = 4 sin x + 7* - 5 (3) Find y' and then evaluate it for (-2,3) e2x-1 (4) Find the limit: lim- x-0 cos x-1 3x³ + y² = -xy (5) Find absolute maximum and absolute minimum of f(x) = x³ 3x² + 1 with the domain of [-2, 4] (6) Compute Vy and dy for f(x) = x - x³ where x = 1 and Vx = 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. Diameter 2 times radius. Circle Area = Circle Area = nr² Sphere Volume. v = (³) Ár³ (8) Find the limit. Must use l'Hospital Rule. In x lim x-xx-1
(1) Find the antiderivative of f(x). Initial condition is f(0) =3. Let f'(x) = 4x³ + x² – 3x (2) Find the function g(x) where g'(x) = 4 sin x + 7* - 5 (3) Find y' and then evaluate it for (-2,3) e2x-1 (4) Find the limit: lim- x-0 cos x-1 3x³ + y² = -xy (5) Find absolute maximum and absolute minimum of f(x) = x³ 3x² + 1 with the domain of [-2, 4] (6) Compute Vy and dy for f(x) = x - x³ where x = 1 and Vx = 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. Diameter 2 times radius. Circle Area = Circle Area = nr² Sphere Volume. v = (³) Ár³ (8) Find the limit. Must use l'Hospital Rule. In x lim x-xx-1
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.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 37E
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![### Calculus Problems and Exercises
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) from \( 3x^3 + y^2 = -xy \)**
4. **Find the limit:**
\[
\lim_{{x \to 0}} \frac{e^{2x} - 1}{\cos x - 1}
\]
5. **Find the absolute maximum and absolute minimum of \( f(x) = x^3 - 3x^2 + 1 \) with the domain of [-2, 4]**
6. **Compute \( \nabla y \) and \( dy \) 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 increasing when the diameter is 26 feet?**
- **Diameter = 2 times radius.**
- **Circle Area = \( \pi r^2 \)**
- **Sphere Volume = \( v = \frac{4}{3} \pi r^3 \)**
8. **Find the limit using 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 the point \( (\frac{\pi}{2}, \pi) \)**
10. **Find the derivative:**
\[
\frac{d}{dt} \left( \frac{\cos t}{1 - \sin t} - e^t (t^3) \right). \quad \text{Evaluate the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b3472e-ab59-40f2-b296-ac3727a4d9f3%2F868d6e28-b577-4050-b080-5d73e5ddaf99%2F4e36pa8i_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problems and Exercises
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) from \( 3x^3 + y^2 = -xy \)**
4. **Find the limit:**
\[
\lim_{{x \to 0}} \frac{e^{2x} - 1}{\cos x - 1}
\]
5. **Find the absolute maximum and absolute minimum of \( f(x) = x^3 - 3x^2 + 1 \) with the domain of [-2, 4]**
6. **Compute \( \nabla y \) and \( dy \) 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 increasing when the diameter is 26 feet?**
- **Diameter = 2 times radius.**
- **Circle Area = \( \pi r^2 \)**
- **Sphere Volume = \( v = \frac{4}{3} \pi r^3 \)**
8. **Find the limit using 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 the point \( (\frac{\pi}{2}, \pi) \)**
10. **Find the derivative:**
\[
\frac{d}{dt} \left( \frac{\cos t}{1 - \sin t} - e^t (t^3) \right). \quad \text{Evaluate the
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