
# Test 3


Exercise $$\PageIndex{1}$$

Calculate the following integrals:

$\displaystyle \int \frac{ x^2+1} {\sqrt{x}} dx$

$\displaystyle \int_{1}^{2} (3x^2+2x-1) \,dx$

$$\dfrac{2 x^{5/2}}{5}+ 2x^{1/2}+C$$, $$9$$

Exercise $$\PageIndex{2}$$

Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval and find all values of $$c$$ in that interval that satisfy the conclusion of the theorem for $$f(x) = x^2-x$$ on $$[-3,5].$$

$$c=1$$

Solution:

Since $$f$$ is a polynomial, it is continuous and differentiable everywhere. Therefore, $$f$$ satisfies the hypotheses of the Mean Value Theorem, and there must exist at least one value $$c∈(-3,5)$$ such that $$f'(c)$$ is equal to the slope of the secant line connecting $$(-3,f(-3))$$ and $$(5,f(5)).$$

To determine which value(s) of $$c$$ are guaranteed, first calculate the derivative of $$f$$. The derivative $$f′(x)= 2x-1$$. The slope of the line connecting $$(-3,f(-3))$$ and $$(5,f(5))$$ is given by

$\frac{f(5)−f(-3)}{5+3}=\frac{20−12}{8}=\frac{8}{8}=1.$

We want to find $$c$$ such that $$f′(c)=\frac{7}{4}$$. That is, we want to find $$c$$ such that

$2c-1=1$

Solving this equation for $$c$$, we obtain $$c=1$$.

Exercise $$\PageIndex{3}$$

Calculate the following integrals by using substitution:

1. $$\displaystyle{\int \left( \frac{1}{\sqrt{t}} - 3\sqrt{t} \right) \, dt}$$.
2. $$\displaystyle{\int 4x\sec^2(x^2)\, dx}$$.
3. $$\displaystyle{\int \sin^4(x)\cos(x) \, dx}$$.
4. $$\displaystyle{\int {1 \over 5x+2} \, dx}$$.
5. $$\displaystyle{\int {\frac{dx}{\sqrt{x}(x+1)}}}$$.
Hint:

2. use $$u= x^2$$,

3. use $$u= \sin(x)$$,

4. use $$u=5 x+2$$,

5. use $$u= \sqrt{x}$$.

1. $$2t^{\frac{1}{2}}-2t^{\frac{3}{2}} +C$$,

2. $$2 \tan(x^2)+C$$,

3. $$\dfrac {\sin^5(x)}{5}+C$$,

4. $$\dfrac {\ln(5x+2)}{5}+C$$,

5. $$2 \tan^{-1}(\sqrt{x})+C$$.

Exercise $$\PageIndex{4}$$

An oil tanker is leaking at the rate given (in barrels per hour) by $L^{'}(t)= \displaystyle \frac{50 \ln(t+1)}{t+1},$ where $t$ is the time (in hours) (when $t=0$). Find the total number of barrels that ship will leak on the end of the first day.

$$25 ( ln(25))^2)$$.

Exercise $$\PageIndex{5}$$

Solve the initial value problem $\frac{dy}{dt} = e^{-2t}, y(0)=5.$

$$y= \dfrac {-e^{-2t}+11}{2}$$.

Exercise $$\PageIndex{6}$$

Find the position, velocity, speed and acceleration at time $$t=1$$, $s(t)=9-9\cos\left( \displaystyle \frac{\pi t}{3}\right), 0 \leq t \leq 5.$

$$s(1)=4.5 ft. v(1) =1.5 \sqrt{3} \pi, a(1)= \pi^2/2.$$

Solution:

$$s(1)=9-9\cos\left( \displaystyle \frac{\pi }{3}\right)= 9-9(\dfrac{1}{2})=4.5 ft$$

$$v(t)=s'(t)= 3 \pi \sin\left( \displaystyle \frac{\pi t}{3}\right)$$, $$v(1)=s'(1)=3 \pi \sin\left( \displaystyle \frac{\pi }{3}\right)= 1.5 \sqrt{3} \pi$$

$$a(t)=v'(t)= \pi ^2 \cos\left( \displaystyle \frac{\pi t}{3}\right)$$, $$a(1)=v'(1)=\pi^2 \cos\left( \displaystyle \frac{\pi }{3}\right)=)= \pi^2/2$$