Math 201 Sample Final Exam 2

Instructions:

This exam contains 10 questions. Students must solve all 10 questions. Total: 100 points. No calculators or electronic devices allowed. Turn off all sound-producing devices. You must show all your work to receive credit. Good luck!

Question 1 (10 points)

Compute $\dfrac{dy}{dx}$ for each function. You do not need to simplify.

(a)
$$y=\frac{x^3-5x^4+x^8}{e^{3x}}$$

(b)
$$y=x^{x+2}$$

(c)
$$xy^3-x^2=e^y-2$$

Question 2 (10 points)

Evaluate each integral.

(a)
$$\int x^5 \sin(x^6)\,dx$$

(b)
$$\int \frac{2^{1/x^2}}{x^3}\,dx$$

(c)
$$\int \frac{e^{(2+\ln(x))}}{x}\,dx$$

Question 3 (10 points)

Find the limit or state that it does not exist. Justify your answer.

(a)
$$\lim_{x\to -\infty}\frac{x+3}{\sqrt{4x^2+5x-8}}$$

(b)
$$\lim_{x\to\infty}\ln(x^{1/x})$$

(c)
$$\lim_{x\to 0} \left[ \frac{1}{\ln(x+1)}-\frac{1}{x} \right]$$

Question 4 (10 points)

(a) State the Fundamental Theorem of Calculus, Part I (include all hypotheses).

(b)
Let
$$F(x)=\int_0^x \frac{t^2}{t^2+t+2}\,dt$$ Find $F''(x)$ and determine the concavity of $F$.

Question 5 (10 points)

Water is leaking from an inverted conical tank at a rate of $100\text{ cm}^3/\text{min}$. The tank has height $6\text{ m}$ and diameter $4\text{ m}$.

If the water level is rising at $20\text{ cm/min}$ when the height of water is $0.3\text{ m}$, find the rate at which water is being pumped into the tank.

Volume of a cone: $$V=\frac{\pi}{3}r^2h,\qquad 1\text{ m}=100\text{ cm}$$

Question 6 (10 points)

Let $$s(t)=t^3-10t^2+25t$$ be the position function.
Find the object’s acceleration each time the speed is zero.

Question 7 (10 points)

An island is $\sqrt{3}$ miles north of the shore. A guard starts 6 miles west of the closest shore point.

Running speed: $4\text{ mi/hr}$
Swimming speed: $2\text{ mi/hr}$

How far should the guard run before swimming to minimize travel time?

Question 8 (10 points)

Let $$f(x)=\frac{1}{2}x^4+\frac{8}{3}x^3-\frac{2}{3}.$$

(a) Where is $f$ increasing/decreasing?
(b) Where does $f$ have local extrema?
(c) Where is $f$ concave up/down?
(d) Does $f$ have inflection points? If so, find them.
(e) Sketch the graph.

Question 9 (10 points)

(a) Use the limit definition of the derivative to find $$f'(x)\quad\text{if}\quad f(x)=\sqrt{x+1}.$$

(b) Find the equation of the tangent line at $x=3$.

(c) Use differentials to estimate $f(3.01)$.

Question 10 (10 points)

(a) Define $$\int_a^b f(x)\,dx$$ in terms of Riemann sums.

(b) Use 4 subintervals and right endpoints to estimate $$\int_1^9 \sqrt{\ln x}\,dx.$$

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