Test 1 review questions
The best way to prepare for this test is to practice on as many problems as possible. If you haven’t done all of the assigned homework from the book, or all of the worksheets, do so now! If you have, make sure you look back over them, so that you remember what you need to do to solve the different types of problems on the test. You can also use the gateway modules for extra practice on the algebra and interpreting limits.
Finally, here are 18 more questions, all of which are similar to questions I have asked in the past.
1. If : Evaluate f(2), f(a), f(2+a), and .
2. The population of Rockville was 47,000 in 2000, and grew to 61,000 in 2010.
a. Assuming the population grew linearly, find a formula for p, the population of the city as a function of t, the number of years since 2000.
b. Interpret the slope of your answer to part (a) in practical terms (ie – your answer should include units and a description of what it tells you in the context of the problem)
The slope of 1400 represents the annual increase in population over the given decade.
c. Now assume that the population of Rockville grew exponentially. Find a formula for p, the population of the city as a function of t, the number of years since 2000.
3. If , find each of the following:
4. Express as a composition of functions
5. For the function f(x) graphed to the right:
a. What is the domain? The range? Domain, [-3,3],
Range [-4 5]
b. Is f(x) invertible? Why or why not?
No, it fails the horizontal line test (or it is not 1-1)
c. Graph f(2x), f(x-2), and f(x)+1
Sketch the graph, 1) compressed horizontally by a factor of 2 (so it runs from -1.5 to 1.5),
2) shifted to the right 2 (so it runs from -1 to 5), and
3) shifted up 1.
6. Use your calculator to find all solutions to . Round your answers to 3 decimal places.
-2.879, -.653, .532
7. If f(x) = , find a formula for f-1(x)
8. For the function shown in the graph to the right (in blue):
a. What is the value of f(2)? The value of f-1(2)?
b. Sketch the inverse of the function on the same set of axes. (see red)
9. Zoom in on the graph of near the point (2, 8) until it looks like a straight line.
Then use the trace feature to find the coordinates of another point on the curve, and find the slope of the line connecting those 2 points. Finally, find an equation for that line.
10. Let P be the point (2, 8). Find the slope of the secant line PQ, where Q is the point for the following values of x (Give answers to 3 decimal places)
a) 3 b) 2.1 c) 2.001
a) 19; b) 12.61; c) 12.006
d) Using parts (a)-(c) above, and additional points if necessary, approximate the slope of the tangent line to at the point x=2.
The limit of the above slopes appears to be 12, so this should be the slope of the tangent line.
e) What is the connection between this problem and #9?
This is a numerical approximation to the slope of the same line that we found graphically in 9.
11. Use the definition of the derivative to find the derivative of at (2,-2). Then use this to find the tangent line to this function at that point
So the tangent line goes through (2,-2) with slope -3, which is y=-3x+4
12. f(x) is the function graphed to the right:
a. Does exist? If so, what is it?
b. Does exist? If so, what is it?
c. Does exist? If so, what is it?
d. Does f(-1) exist? If so, what is it?
e. Is f continuous at x= -1? Why or why not?
No, the limit is not the same as the value there
f. Is f continuous at x= 1? Why or why not?
Yes, the limit is the same as the value there, 0
13. Evaluate the
following limits. Use numeric and
graphic means, and where possible also use algebra to confirm the answer:
Only the algebraic solutions are shown for the first 2. You should approximate these numerically and graphically too:
This is not a proof, but reason to at least suspect that the limit is e, which it is.
DNE. If x>2, the fraction goes to ∞, but if x<2 it goes to -∞.
14. If for all x, , what is ?
2, by the squeeze theorem.
15. Find numbers a and b such that is continuous everywhere.
For this to be continuous at x=2, the limit from the left, limit from the right, and the value must all agree. From the left the limit is 4. So a=4, and b=6.
16. Carefully explain why there must be a number x between 0 and 1 such that
If x=0, x3+3x2=0, while if x=1, x3+3x2=4. Since f(x)= x3+3x2 is a continuous function, the IVT applies, and states that between x=0 and x=1 f(x) must hit every value between 0 and 4 (including 1).
17. The height of a ball, t seconds after it is thrown, is given by .
a. What is the average speed of the ball over the first second after it is thrown?
b. What is the instantaneous speed of the ball one second after it is thrown?
18. The graph shows f, f’, and f’’. Which curve is which? How do you know?
The blue is f, its derivative f’ is in red, and red’s derivative is f’’ in green.
When the blue curve is increasing the red curve is positive, and when it is decreasing the red is negative. Also, when the blue is concave down, the red (its derivative) is decreasing and the green (its 2nd derivative) is negative, while when the blue is concave up the red is increasing and the green is positive.