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You CAN Ace Calculus

17calculus > exam list > calc1 exam A1

 limits derivatives Most limit topics are required for this exam except for L'Hôpital's Rule which is usually covered in calculus 2.

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

related topics

exam list

Calculus 1 - Exam A1

This is the first exam for first semester single variable calculus.

Exam Details

Tools

Time

1.5 hours

Calculators

see instructions

Questions

17

Formula Sheet(s)

none

Total Points

100

Other Tools

none

Calculus 1 - Exam A1

This is the first exam for first semester single variable calculus.

Exam Details

Time

1.5 hours

Questions

17

Total Points

100

Tools

Calculators

see instructions

Formula Sheet(s)

none

Other Tools

none

Instructions:
- This exam is in two main parts, labeled parts A and B, with different instructions for each part.
- 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).

#### Part A - Questions 1-11

Instructions for Part A - - You have 30 minutes to complete this part of the exam. No calculators are allowed. All questions in this section are worth 3 points, except for question 10, which is worth 4 points.

Section 1

Find the indicated limit or determine that it doesn't exist.

Question 1

$$\displaystyle{ \lim_{\theta \to 0}{\left[ \frac{\sin(\theta)}{\tan(3\theta)} \right]} }$$

solution

Question 2

$$\displaystyle{ \lim_{x \to 2}{\left[ \frac{x^2+x-6}{x-2} \right]} }$$

solution

Question 3

$$\displaystyle{ \lim_{x \to 3}{\left[ \frac{x^2-4x+6}{x^2-9} \right]} }$$

solution

Question 4

$$\displaystyle{ \lim_{x \to 1}{\left[ \frac{x-1}{x-x^3} \right]} }$$

solution

Question 5

$$\displaystyle{ \lim_{x \to \infty}{\left[ \frac{5x-1}{3x^2+1/4} \right]} }$$

solution

Section 2

Evaluate the following derivatives and simplify. Give all exact answers.

Question 6

$$\displaystyle{ \frac{d}{dx} \left[ (3x^2+x+1)^{2010} \right] }$$

solution

Question 7

$$\displaystyle{ \frac{d}{dx}\left[ \frac{\sin^2 (x)}{\cos(x)} \right] }$$

solution

Question 8

Find $$D_x y$$ for $$y = \sec^2 (4x)$$

solution

Question 9

Find $$f'(3)$$ for $$f(x) = (2+x^2)^{4/3}$$

solution

Question 10

Find $$dy/dx$$ using implicit differentiation
for the curve $$2x-y^2 = \cos(xy)+5$$

solution

Question 11

Find $$D_x^{(101)} [ \cos(x) ]$$

solution

#### Part B - Questions 12-17

Instructions for Part B - - You may use your calculator. You have one hour to complete this part of the exam. Show all your work. Correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions. Give exact answers.

Section 3

Solve the following problems.

Question 12

[12 points] Find the equation of the tangent line to $$\displaystyle{ y = \frac{1}{x^2+4} }$$ at $$x=2$$. Give your answer in slope-intercept form.

solution

Question 13

[12 points] Sketch the graph of a function $$g(x)$$ that satisfies the following conditions. Be sure to label your graph and axes.
(a) $$g(x)$$ is continuous at $$x=1$$     (b) $$g'(1)$$ does not exist
(c) $$g(1) = g(2) = 3$$     (d) $$\displaystyle{ \lim_{x \to 2^+}{g(x)} = -\infty }$$     (e) $$\displaystyle{ \lim_{x \to \infty}{g(x)} = 1 }$$

solution

Question 14

[12 points] Identify all critical points and find the maximum and minimum value of the function
$$f(x) = x^3-3x^2+1$$ on $$S = [-1,4]$$.

solution

Question 15

[15 points] A ladder 3 meters long, with one end on the ground, is raised so that the angle $$\theta$$ of the ladder with the ground increases by 2.5 radians per minute. Consider the right triangle formed by the ladder, the ground, and a vertical line through the top of the ladder. How fast is the area of this triangle increasing at the time when the angle $$\theta$$ equals $$\pi/6$$?

solution

Section 4

For the questions in this section, consider the function $$f(x) = x^4-4x^3-10$$.

Question 16

[8 points] Find the intervals on which $$f(x)$$ is decreasing and the intervals where $$f(x)$$ is increasing.

[7 points] Find the intervals on which $$f(x)$$ is concave up and the intervals where $$f(x)$$ is concave down.