Question 1

Evaluate the integral $$\displaystyle \$$int_0$$^1 \frac{x-7}{x^2$ - 2x - 3} \, dx.$$

Accepted Answer

$$\displaystyle -2 \ln 2$$

Question 2

Compute the integral $$\displaystyle \$$int_0$$^2 \frac{x^2$}{4 + x^2$} \, dx.$$

Accepted Answer

$$\displaystyle 2 - 4 \arctan\left(\frac{2}{2}\right) = 2 - 4 \arctan(1) = 2 - 4 \frac{\pi}{4} = 2 - \pi$$

Question 3

Let $$f(x) = (x^2$ - 4x + 4) e^x. $$Determine $$\displaystyle \lim_{x 	o -\infty} f(x)$$ and $$\displaystyle \lim_{x 	o \infty} f(x).$$

Accepted Answer

$$0$$ and $$\infty$$

Question 4

For $$f(x) = (x^2$ - 4x + 4) e^x$$, on which intervals is $$f$$ increasing?

Accepted Answer

$$x \u003e 3$$

Question 5

Find all local extrema of $$f(x) = (x^2$ - 4x + 4) e^x.$$

Accepted Answer

Local minimum at $$x=2$$, no local maximum

Question 6

Find the second-degree Taylor polynomial for $$f(x) = \sqrt{x}$$ centered at $$x=25.$$

Accepted Answer

$$\displaystyle 5 + \frac{1}{10}(x-25) - \frac{1}{200}(x-25)^2$

Question 7

Use the Taylor polynomial from the previous question to approximate $$\sqrt{27}.$$

Accepted Answer

$$\displaystyle 5.095$$

Question 8

Can you be certain that the error in the approximation of $$\sqrt{27}$$ using the degree 2 Taylor polynomial at $$x=25 is $$less than $$\frac{1}{5000}$$?

Accepted Answer

No, because the error bound exceeds $$\frac{1}{5000}.$$

Question 9

Evaluate the improper integral $$\displaystyle \$$int_1$$^\infty \frac{x}{e^x} \, dx.$$

Accepted Answer

$$\displaystyle 2e$$^{-1}$$

Question 10

Determine whether the improper integral $$\displaystyle \$$int_1$$^\infty \frac{x+1}{2e^x + \cos x} \, dx$$ converges.

Accepted Answer

Yes, because the denominator grows faster than the numerator.

Question 11

The region bounded by $$x^{2/3}$$ + $$y^{2/3} = 1$ in the first quadrant is rotated about the x$-axis. What is the volume of the resulting solid?

Accepted Answer

$$\displaystyle \frac{3\pi}{4}$$

Question 12

Given the astroid curve parametrized by $$x = \cos^3$ t$$, $$y = \sin^3$ t$$ for $$0 \leq t \leq \frac{\pi}{2}$$, what is the length of the curve in the first quadrant?

Accepted Answer

$$\displaystyle \frac{3}{2}$$

Question 13

Determine whether the series $$\displaystyle \sum_{k=3}^\infty \frac{1}{k \ln k}$$ converges.

Accepted Answer

Diverges, by comparison to the harmonic series.

Question 14

Does the series $$\displaystyle \sum_{k=1}^\infty \frac{1}{k + \ln(k+1)}$$ converge?

Accepted Answer

No, it diverges because the denominator grows too slowly.

Question 15

Suppose a medication's concentration in the bloodstream is modeled by $$C(t) = (t^2$ - 4t + 4) e^t$$, where $$t is $$time in hours. At what time is the concentration minimized?

Accepted Answer

$$t = 2$$ hours

Question 16

A tank is being filled such that the rate of water inflow at time $$t is$$ $$r(t) = \frac{t}{e^t}$$ liters per minute. How much water enters the tank from $$t=1 to$$ $$t=\infty$$?

Accepted Answer

$$2e^{-1}$$ liters

Question 17

Which of the following best describes the behavior of $$f(x) = (x^2$ - 4x + 4) e^x as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x)$$ approaches $$\infty$$

Step-by-Step Calculus Exam Guidance: SF1625 Envariabelanalys

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