A differentiable function is measured at integer inputs
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A differentiable function q is measured at integer inputs, producing the data below.
Numerical data (no units):
xq(x)
0 5.2
1 4.7
2 3.9
3 4.2
4 5.1
(a) Using only the values in the table, compute an estimate for q′(x) at each integer x=1,2,3. Show each calculation clearly.
(b) Based on your results from part (a), identify the intervals on which q appears to be increasing and those on which it appears to be decreasing. Justify your answer numerically.
(c) Sketch a rough graph of q′ on the integer interval [1,3] using your numerical estimates. Explain how the shape of this sketch provides evidence for a local extremum of q and state the approximate location of that extremum.
(d) Suppose an additional measurement at x=2.5 revealed q(2.5)=3.7. Discuss how incorporating this new data point would alter at least one of the derivative estimates you computed and the conclusions you drew in parts (b) and (c).