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Math 1100

Exam 1 Review

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MATH1100: Calculus I Test #1 Review Problems To get the most benefit, you should attempt all of the problems below without a calculator and without reference to your notes, the textbook, or any online resources. Do not look at the solutions until you have attempted the problem. Disclaimer: The problems below represent some types of problem that will be asked on the exam, but are not intended to be an exhaustive list of all possible topics or problem types on the exam. You are responsible for any topics covered during class, video learning modules, labs and on homework, both online and written. 1. Find the domain of the following functions: sin x (a) f(x) = 1 − cos2 x p

(b) g(x) = ln

x2 − 4 + 1



2. Consider the piecewise function defined as follows:

f(x) =

 2   x + 5x + 6    x+2 

1

−    x−1   

ln(x − 1)

(a) Find the domain.

for −3 ≤ x ≤ −1, for −1 < x < 1, for 1 < x ≤ 2.

(b) Find the numbers at which f is discontinuous and classify each discontinuity (removable, jump, or infinite). All work must be justified using limits.

3. Find the average rate of change of the following functions on the given intervals. Then sketch a rough graph of the function and indicate how to visualize the answers you computed. (a) f(x) = x3 + 1 i. [2, 3] ii. [−1, 1] (b) g(t) = 2 + cos t i. [0, π] ii. [−π, π]

(Remember, DO NOT use a calculator.)

Math 1100

Exam 1 Review

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4. The graph below show the total distance s traveled by a bicyclist after t hours.

(a) Estimate the bicyclist’s average speed over the time intervals i. [0, 1] ii. [1, 2.5] iii. [2.5, 3.5] (b) Estimate the bicyclist’s instantaneous speed at times t = 0.5, t = 2, t = 3. Justify your answer with a calculation or using the graph. (c) Estimate the bicyclist’s maximum speed and the specific time at which it occurs. Justify your answer with a calculation or using the graph. 5. A function is given below by a table of values. x f(x)

1 4

2 6

3 2

(a) Fill out the following table of values of f ◦ f . 1

x (f ◦ f )(x)

4 5 2

5 1 3

6 3 4

5

6

(b) Fill out the following table of values of f −1 . x f −1 (x)

6. Let f(x) =

1

2

3

4

5

6

1 + 4x . Find f −1 (x). 5 + 3x

7. Let f(x) = e2x+5. Find f −1 (x). 8. You and a friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 3, and then add 4, before telling you the final answer. (a) If your friend chose the number x, what is the equation for the final answer f(x)? (b) Find f −1 . What does f −1 represent in this context? (c) If your friend’s final answer is 79, what was the number chosen?

Math 1100

Exam 1 Review

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9. The table below gives the average weight, w, in pounds, of American men in their sixties for height, h, in inches. h w

68 166

69 171

70 176

71 181

72 186

73 191

74 196

75 201

(a) How do you know that the data in this table could represent a linear function? (b) Find the weight w as a linear function of height h. What is the slope of the line? What are the units of the slope? (c) Find the height h as a linear function of weight w. What is the slope of the line? What are the units of the slope?

10. According to Mark Zuckerburg, the founder of facebook.com, the amount of personal information which people are willing to put on the internet grows exponentially. (This is known as Zuckerburg’s law.) Suppose that on January 1st 2004, there are (in total) 2 billion bytes of personal information on the internet. By January 1st 2006, there are 30 billion bytes. For this question, we will assume that Zuckerburg’s law is true. (a) Let f(t) be the function which gives the number of billions of bytes of personal information on the internet t years after January 1st 2004. (So f(0) = 2.) Work out a formula for f(t). (b) What is percent increase of information on the internet every two years? Explain briefly. (c) What was the average rate of change of personal information on the internet between 2004 and 2006? Include units.

11. A petri dish is plated with 60 cells that grow at a rate of 15% per day. (a) Give a formula for the number of cells in the dish as a function of the time in days since the cells were plated. (b) Give a formula for the number of days since the cells were plated as a function of the number of cells in the dish.

12. Find the value of a so that lim f(x) exists, where f (x) = x→2

reasoning completely, using limits.

(

x2 + 1

for x > 2,

ax − 3

for x ≤ 2.

Explain your

Math 1100

Exam 1 Review

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13. Calculate the following limits. Remember to show your work, simplify as much as possible, and DO NOT use a calculator! If the limit does not exist, explain why and state whether it is ∞, −∞ or neither. x2 − 5x + 6 (a) lim 2 x→2 x − x + 2 x−3 (b) lim √ x→3 x2 − 9 2 cos x (c) lim x x→π/2 (d) lim

x→0

|x| x

x2 − 4 x→2 x2 − 4x − 4 √ t2 + 1 − 1 (f) lim t→0 t

(e) lim

1 + 3x . 14. Let f(x) = √ 2 x −4 (a) Give the domain of the function in interval notation. (b) Find all vertical asymptotes, if any. Justify your answer using limits. (Answers without limit justification will receive little or no points.)

15. A function y = f(x) is given by the following graph:

(a) Is the function invertible? Why or why not? (b) Find the limits at x = −2 and x = 0. (c) Give the equation of all vertical asymptotes, if any. (d) Sketch a possible continuation of the graph assuming that lim = −∞ and lim = 2. If x→∞

it is not possible, explain why.

x→−∞

Math 1100

Exam 1 Review

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16. Give an example of a function with a vertical asymptote at x = 2, a removable discontinuity at x = 3 and no horizontal asymptotes. Create a specific formula for your function, not just the graph. If it is not possible, explain why.

(

x + 2, x ≤ 0 and g(x) = 2x, does lim f(x)g (x) exist? Explain. If you use any x→0 −3, x>0 limit laws, clearly state these.

17. If f(x) =

18. Suppose we have a rectangle R whose side lengths are given by a(x) = x2 +3 and b(x) = What is the area of the rectangle as x gets infinitely large?

1 . 5x2 +x+3

19. Calculate the following limits, if they exist. Remember to show your work, simplify as much as possible, and DO NOT use a calculator! If the limit does not exist, explain the behavior of the function and why the limit does not exist. 3 x4 − 3 x + 2 (a) lim x→−∞ 5x4 − 10x3 + 2x2 (b) lim

x→∞

(c)

1 + ex 2e1/x

lim cos(x)

x→−∞

(d) lim ln(x) x→∞

(e) lim arctan(x) x→∞

(f)

lim arctan(x)

x→−∞



(g) lim x + x→∞

p

x2 + x + 1



20. Find all horizontal and vertical asymptotes of each function. Calculate all relevant limits. 5 + 4x (a) y = x+3 (b) y = (c) y =

2x2 + x + 1 x2 + x − 2 2ex −5

ex...