MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The functions h and k are defined for x near 4. Portions of their tables are shown; one value of h is intentionally left blank.
Table 1: h(x)
x
3.90
3.99
3.999
4.001
4.01
4.10
h(x)
7.80
7.98
\_\_\_\_
8.002
8.02
8.20
Table 2: k(x)
x
3.90
3.99
3.999
4.001
4.01
4.10
k(x)
−0.10
−0.01
−0.001
0.001
0.01
0.10
Assume both functions are differentiable on an open interval containing x=4.
(a) Use the apparent numerical pattern in Table 1 to supply a plausible estimate for h(3.999).
(b) With the completed table, estimate x→4limk(x)h(x)−8. Show all numerical computations used to support your estimate.
(c) Explain how the limit evaluated in part (b) is related to the derivative of h at x=4. State any assumptions you make explicit in your reasoning.