M160 Midterm Exam 1 - Open Computing Facility



1. (8 points) The height (in feet) of a small weight oscillating at the end of a spring is h(t) = 0.5 cos(2t)

(t in seconds, 0 < t < 5). (This is calculus. Use radian mode!)

(a) Calculate the weight’s average velocity over the time interval 1 < t < 3. Show clearly how you calculated the average velocity. (Round your answer to 5 decimal places.)

(b) Calculate the weight’s average velocity over the time interval 0 < t < π. Show clearly how you calculated the average velocity. (Exact values required!)

Explain how this result can be right when the weight is obviously moving all the time!

(c) Explain why the instantaneous velocity of the weight at the instant t = 2 seconds cannot be found using only arithmetic and algebra.

(d) Use the connection between average velocity and instantaneous velocity to determine whether the instantaneous velocity of the weight at time t = 2 seconds is greater than or less than 0.76 feet/sec.

Explain clearly how you decided.

(Your explanation must not use the fact that the derivative of the cosine function is the sine function!)

2. (8 points) We know that [pic] = 1. This means that that for every ε > 0 there exists a corresponding δ > 0 such that for all x in the domain of f if 0 < |x – 2| < δ, then [pic] < ε.

Sketch a relevant graph and use it (with your calculator) to find a numerical value for δ when

ε = 0.2 in the limit statement [pic] = 1.

δ = ________________

In your sketch illustrate clearly how you found δ from the graph.

Clearly label the units on the vertical axis.

Show clearly how you found δ from the graph. No credit for only algebraic work.

| | | x

1 2 3

3. (8 points) (a) State the mathematical definition of the phrase “a function y = f(x) is continuous at x = c”.

Let g(x) be the function defined by

g(x) = [pic]

(b) Is the function g(x) continuous at the point c = 2? Show how your conclusion follows from the mathematical definition of a function being continuous at a point (from (a)).

(c) Is the function g(x) continuous at the point c = 1? Show how your conclusion follows from the mathematical definition of a function being continuous at a point (from (a)).

4. (10 points)

(a) Find the slope of the line tangent to the graph of y = [pic] at the point (1, 1) by using the definition of the slope as the limit of slopes of secant lines. Evaluate the limit algebraically. Show the details of the algebra.

(b) Write an equation in point-slope form for the line tangent to the graph of y = [pic] at the point (1, 1) .

5. (10 points) LeAnn and Aaron are studying MATH 160 together. “Look,” LeAnn says to her classmate, “it says here in the Study Guide that we should be able to explain in common, non-technical language what it means when they say that a function y = f(x) has a finite number L as its limit as x approaches a finite number a. I’ll bet they ask us to do that on the midterm exam next week.”

“That’s a no-brainer,” Aaron replies confidently. “It just means that as you choose x-values closer and closer to a, the values of the function f(x) are closer and closer to L.”

LeAnn isn’t convinced. She suggests they look at some examples together.

(a) LeAnn’s first example:

(i) What is the limit of the function f(x) = –x2 as x approaches the number a = 0? ____________

(ii) Find all numbers L so that the function f(x) = –x2, the number a = 0, and the number L are related in the way Aaron describes. Sketch a graph and use it to explain how to see that the values of the function f(x) = –x2, the number a = 0, and all the numbers L you found are related in the way Aaron describes.

(b) LeAnn’s second example:

(i) What is the limit of the function f(x) = xcos[pic] as x approaches a = 0? L = ______________

(ii) Sketch a graph and use it to explain how to see that the values of the function f(x) = xcos[pic], the number a = 0, and the limit L you found are not related in the way Aaron describes.

(c) What could LeAnn say to Aaron to convince him that her examples show that his non-technical explanation of what it means to say that a function y = f(x) has limit L as x approaches a is not accurate?

(d) Write a sentence that explains in every-day non-technical terms what it means when we say that a function

y = f(x) has a finite number L as its limit as x approaches a finite number a.

(“f(x) approaches L as x approaches a” says the same thing as Aaron said.)

6. (30 points) Evaluate the following limits. Include the possibility that the limit might not exist or the limit may be infinite. In each case, show the essential algebraic steps. If a limit is infinite or does not exist, say so and explain how you reached that conclusion. Use the back of the facing page if you need more space.

(a) [pic][pic] =

(b) [pic][pic] =

(c) [pic][pic] =

(d) [pic][pic] =

(e) [pic][pic] =

(f) [pic][pic] =

7. (10 points) Sketch the graph of a function y = f(x) that has all the following properties:

▪ [pic]f(x) = – 3

▪ [pic] f(x) = – ∞

▪ [pic]f(x) = + ∞

▪ [pic] f(x) = 1

▪ [pic] f(x) = – 1

▪ [pic] f(x) = 4

Mark and label the units on the vertical axis clearly!

y

–

–

–

1 –

| | | | | | | x

-3 -2 -1 0 1 2 3 4

-1 –

–

–

–

(b) Complete the sentence by filling in the blanks with a two-word phrase.

The lines y = – 3 and y = 4 are ____________________ __________________ of the graph of this function.

(c) Complete the sentence by filling in the blanks with a two-word phrase.

The line x = – 1 is a ____________________ __________________ of the graph of this function.

(d) Does the phrase you used to fill in the blank in (c) also describe the line x = 2? Explain why or why not.

8. (8 points) Let h(x) = [pic] . Use the Intermediate Value Theorem to explain how to see without using a calculator and without drawing a graph that there is a number c where h(c) = [pic].

(Give a complete explanation. Use complete sentences.)

9. (8 points) Suppose you know that [pic]f(x) = L and [pic]f(x) = M where L and M are numbers.

(L and M are not ±∞ because +∞ and -∞ are not numbers.)

(a) Can you conclude that limx→1 f(x) does not exist? YES NO

Explain why or why not. If you wish, you may use graphs to illustrate your explanations.

(b) Can you conclude that f(1) is not defined? YES NO

Explain why or why not. If you wish, you may use graphs to illustrate your explanations.

(c) Can you conclude that the graph of f has a vertical asymptote at x = 1? YES NO

Explain why or why not. If you wish, you may use graphs to illustrate your explanations.

(d) Can you conclude that the function y = f(x) is continuous at the point c = 1? YES NO

Explain why or why not. If you wish, you may use graphs to illustrate your explanations.

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