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Math 131

Midterm 1 Practice/Review

        Disclaimer: This sheet is excellent practice, but it doesn’t cover all topics that might ap-pear on the midterm and it is certainly not the midterm with different numbers. Look over the practice worksheets! Also check the Essentials.pdf for a list of standard problems... you can expect the midterm to be a tour of these.

1. Sketch graphs for Nick’s position, velocity, and acceleration:

Nick starts jogging and runs faster and faster for 3 minutes, then he walks for 5 minutes. Then Nick waits at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.

2. Let f(x) = log5(x) − 6 and g(x) = x4.

(a) Find f ○ g and give its domain in interval notation.

(b) At what x-value is f(x) = −4.

(c) Decide if g(x) is even/odd/neither by checking the definition.

(d) Is f(x) one-to-one? If so, find f-1(x); if not, explain why not.

(e) Write a function for h(x), the function whose graph is that of y = g(x) flipped across the x-axis and shifted left 5 units.

(f) Describe the graph of the function h(x) = f(−x) + 4 in terms of the graph of f.

3. Calculate the following limits. Good work counts!

(a) 

(b) 

(c) 

(d) 

4. (Good challenge) Find the domain of the function:

5. Solve the given equation for x and simplify:

(a)

(b) 

6. Does the following limit exist? Explain your reasoning.

7. Let

What value of c makes f(x) continuous?

8. Calculate the derivative 

9. Find the equations of all asymptotes of the function

10. Explain why ex sin x = cos x has a solution in the interval 

11. A car’s distance in miles along a straight road is given by where t is hours after 12pm.

(a) Interpret in the context of the problem.

(b) Interpret in the context of the problem.

[Note: You aren’t asked to calculate anything]

12. What are the three types of discontinuities described in class? What part of the definition for continuity at a point does each fail? For each type of discontinuity, sketch a function whose domain is all reals and whose derivative has the discontinuity.