4.0E: Exercises

Exercise $\PageIndex{1}$

1. Define the term “antiderivative” in your own words

2. Is it more accurate to refer to “the” antiderivative of $f(x)$ or “an” antiderivative of $f(x)$?

3. Use your own words to define the indefinite integral of $f(x)$.

4. Fill in the blanks: “Inverse operations do the ____ things in the _____ order.”

5. What is an “initial value problem”?

6. The derivative of a position function is a _____ function.

7. The antiderivative of an acceleration function is a ______ function.

Answer

Under Construction

Exercise $\PageIndex{2}$

Evaluate the indefinite integrals:

1. $\int 3x^3 \,dx$

2. $\int x^8 \,dx$

3. $\int (10x^2-2) \,dx$

4. $\int \,dt$

5. $\int 1 \,ds$

6. $\int \frac{1}{3t^2}\, dt$

7. $\int \frac{1}{t^2}\, dt$

8. $\int \frac{1}{\sqrt{x}}\, dx$

9. $\int \sec^2 \theta\, d\theta$

10. $\int \sin \theta\, d\theta$

11. $\int (\sec x \tan x +\csc x \cot x )\, dx$

12. $\int 5e^\theta\, d\theta$

13. $\int 3^t\, dt$

14. $\int \frac{5^t}{2}\, dt$

15. $\int (2t+3)^2\, dt$

16. $\int (t^2+3)(t^3-2t)\, dt$

17. $\int x^2x^3\, dx$

18. $\int e^\pi\, dx$

19. $\int a\, dx$

Answer

Under Construction

Exercise $\PageIndex{3}$

This problem investigates why Theorem 35 states that $\int \frac{1}{x}\,dx = \ln |x|+C$.
(a) What is the domain of $y=\ln x$?
(b) Find $\frac{d}{dx}(\ln x)$.
(c) What is the domain of $y=\ln (-x)$?
(d) Find $\frac{d}{dx}\left ( (\ln (-x)\right )$.
(e) You should find that $1/x$ has two types of antiderivatives, depending on whether $x>0$ or $x<0$. In one expression, give a formula for $\int \frac{1}{x}\,dx$ that takes these different domains into account, and explain your answer.

Answer

Under Construction

Exercise $\PageIndex{4}$

Find $f(x)$ described by the given initial value problem.

1. $f'(x)=\sin x\text{ and }f(0)=2$

2. $f'(x)=5e^x\text{ and }f(0)=10$

3. $f'(x)=4x^3-3x^2\text{ and }f(-1)=9$

4. $f'(x)=\sec^2 x\text{ and }f(\pi/4)=5$

5. $f'(x)=7^x\text{ and }f(2)=1$

6. $f''(x)=5\text{ and }f'(0)=7,f(0)=3$

7. $f''(x)=7x\text{ and }f'(1)=-1,f(1)=10$

8. $f''(x)=5e^x\text{ and }f'(0)=3,f(0)=5$

9. $f''(\theta)=\sin \theta \text{ and }f'(\pi)=2,f(\pi)=4$

10. $f''(x)=24x^2+2^x-\cos x \text{ and }f'(0)=5,f(0)=0$

11. $f''(x)=0\text{ and }f'(1)=3,f(1)=1$

Answer

Under Construction

Exercise $\PageIndex{5}$

Use information gained from the first and second derivative to sketch $f(x)=\frac{1}{e^x+1}$.

Answer

Under Construction

Exercise $\PageIndex{6}$

Given $y=x^2e^x\cos x$, find $dy$.

Answer

Under Construction