Values of a continuous function are given in the table below
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Values of a continuous function f are given in the table below. The function g is defined for 0≤x≤5 by g(x)=∫0xf(t)dt.
x
0
1
2
3
4
5
f(x)
2.0
3.0
3.8
3.9
3.5
2.8
(a) Use the data to estimate f′(x) on each unit sub-interval [k,k+1] for k=0,1,2,3,4. State the numerical estimate and its sign for each interval.
(b) For each sub-interval in part (a), decide whether g is concave up, concave down, or whether the concavity cannot be determined from the data provided. Summarize your results in a sentence.
(c) Explain how the signs of your estimates in part (a) support your conclusions about the concavity of g in part (b).