First Nine Weeks Mid-term Exam
[AP Calculus AB]
Name: ______
Date: ______Block: 1 2 3 4
Warm-up Q&A
i. To find the slope of the tangent line to a function at any x, first calculate the ______, then evaluate that (or f’(x)) at the specified point or x-value.
ii. Complete the definition of derivative.
f’(x) =
Show all work. Box in final answers clearly. Use extra paper as needed and include with exam.
1. Determine if y = 3x4 + x2 has symmetry to the x-axis, y-axis, or origin.
2. Write the equation of a graph that has intercepts x = and is symmetric with respect to the y-axis.
3. Given f(x) = 4 – x2 and g(x) = 3x + 1, find g(f(x)).
4. Sketch the graph of f(x) = 7 cos x.
Evaluate the following limits:
5.
6.
7.
Use the graph of f(x) to evaluate each limit.
8. 9.
10. Find the slope of the tangent line to the graph of f(x) = x3 + 2x2 at the point when x = 1.
Find the derivative of the following functions using the limit process:
11. g(x) = 2x + 5
12. y = 3x2 – 1
13. Find an equation for the tangent line to the graph of f(x) = at the point where x = 2.
14. Sketch the graph of y = 2x - 3 and its derivative, y’. Label each graphed function.
15. When will the graph from #14 have a horizontal tangent line?
16. Sketch the graph of f(x) = x2 – 5 and its derivative, f’(x). Label each graphed function.
17. When will the graph of f(x) in #16 have a horizontal tangent line?
Give the intervals where the following function, h(x), is differentiable.
18. h(x) = -8x2 + 2
19. Is this h(x) continuous and differentiable for all real x-values? Why or why not?
Give the intervals where the following function, g(x), is differentiable.
20. g(x) = |x – 5|
21. Is this g(x) continuous and differentiable for all real x-values? Why or why not?
22. Determine if f(x) is differentiable at x = 1.
f(x) =
Use f(x) from 22 for the following two questions.
23. [derivative from the left, c=1]
24. [derivative from the right, c=1]
25. Determine if f(x) is differentiable at x = 3.
f(x) =
Use f(x) from 22 for the following two questions.
23. [derivative from the left, c=3]
24. [derivative from the right, c=3]
28.
BONUS (5 points):
Find an equation for the tangent line to the graph of f(x) = at the point (1, 1).