Test 1 review questions
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)
c. Now assume that the population of Rockville grew exponentially. Again, 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:
What is the domain? The range?
6. 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.
7. 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
d) Using parts (a)-(c) above, and additional points if necessary, approximate the slope of the tangent line to at the point x=2.
e) What is the connection between this problem and #9?
8. Use the definition of the derivative (either form of the formula, but not a shortcut) to find the derivative of at (2,-2). Then use this to find the tangent line function at that point
9. f(x) is the function graphed to the right:
b. Does f(-1) exist? If so, what is it?
c. Is f continuous at x= -1? Why or why not?
e. Does f(1) exist? If so, what is it?
f. Is f continuous at x= 1? Why or why not?
10. Evaluate the
following limits. Use numeric and
graphic means, and where possible also use algebra to confirm the answer:
11. Find a number b such that is continuous everywhere.
12. 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?
13. The graph shows f, f’, and f’’. Which curve is which? How do you know?