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You CAN Ace Calculus

17calculus > exam list > calc1 exam B3

 derivatives integrals volume integrals

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

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exam list

Calculus 1 - Exam B3 (Final Exam)

This is the final exam for first semester single variable calculus. The topics covered are integration by substitution, both indefinite and definite, solution of separable differential equations, optimization, area between curves, exponential growth/decay and volumes of revolution using both the disc/washer method and the shell/cylinder method.

Exam Details

Tools

Time

2.5 hours

Calculators

not allowed

Questions

10

Formula Sheet(s)

one page

Total Points

135

Other Tools

ruler

Calculus 1 - Exam B3

This is the final exam for first semester single variable calculus. The topics covered are integration by substitution, both indefinite and definite, solution of separable differential equations, optimization, area between curves, exponential growth/decay and volumes of revolution using both the disc/washer method and the shell/cylinder method.

Exam Details

Time

2.5 hours

Questions

10

Total Points

135

Tools

Calculators

not allowed

Formula Sheet(s)

one page

Other Tools

ruler

Instructions:
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).

Question 1

[10 points] Evaluate $$\displaystyle{ \int{ \frac{e^{1/x}}{x^2}~dx } }$$

solution

Question 2

[10 points] Evaluate $$\displaystyle{ \int_{1}^{e^3}{ \frac{\ln\sqrt{x}}{x}~dx } }$$

solution

Question 3

[15 points] Evaluate $$\displaystyle{ \int{ x\sqrt{2x+3}~dx } }$$

solution

Question 4

[10 points] Evaluate $$\displaystyle{ \int_2^5{ x^2\sqrt{x^3-4}~dx } }$$

solution

Question 5

[10 points] $$\int{ \text{_________}~dx} = (ax^2-a^2)^4 + C$$
If $$a$$ is a constant, fill in the blank.

solution

Question 6

[20 points] Find the particular solution of the differential equation $$f''(x)=e^{3x}+\cos(x)$$ that satisfies the initial conditions $$f(0)=10/9$$ and $$f'(0)=4/3$$.

solution

Question 7

[15 points] For the region bounded by the graphs $$y=9-x^2$$, $$x=2$$ and $$x=3$$ in the first quadrant, a. accurately sketch the region; b. using calculus, calculate the area of the region that you sketched in part a.

solution

Question 8

[10 points] The mass of radioactive material in a sample has decreased by $$30%$$ since the decay began. Assuming a half-life of $$1500$$ years, how long ago did the decay begin?

solution

Question 9

[10 points] A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure. If $$85$$m of new fence is available, what dimensions of the pen will maximize its area? No fence will be used along the walls of the animal house.

solution

Question 10

[25 points] The region bounded by the graphs of $$y=2x$$, $$y=6-x$$ and $$y=0$$ is revolved about the line $$y=-2$$ to produce a volume. Calculate the volume that is produced. Set up the integrals for both the disc/washer and shell/cylinder methods and evaluate one of them.
Notes: Make sure you plot and shade the region and show the example rectangles, as well as the distances R, r, p and h. Each volume should have its own plot, i.e. you should have one plot for the disc/washer method and another separate plot for the shell/cylinder method.
[10 points extra credit] Evaluate the other integral.