AP Calc Notes: DI – 8 Functions Defined as Integrals from Tables and Graphs
x
Ex: Let g ( x ) = ∫ f (t )dt where the function f is graphed at right.
1
f
(
1. Evaluate the following:
a. g(1)
t
b. g(2)
c. g(4)
d. g(7)
e. g(10)
f. g(0)
2. Answer and justify the following:
Where is g increasing?
Decreasing?
f
(
Does g have any relative extrema, if so, where?
What is the maximum value for g on [0, 10]?
The minimum value?
Concave up?
Concave down?
Does g have any points of inflection, if so, where?
What is g(4) – g(2)? Sketch what this means on the graph.
t
Ex: Let y(t) represent the temperature of a pie that has been removed from a 450°F oven and left to cool in a
room with a temperature of 72°F, where y is a differentiable function of t. The table below shows the
temperature recorded every five minutes.
t (min)
0
5
10
15
20
25
30
y(t) (°F)
450
388
338
292
257
226
200
Find an approximation for y ' (18) and explain what it means, indicate units.
25
Find the value of
∫ y ' ( t )dt
and explain what it means, indicate units.
10
25
Find the value of
1
y ' ( t )dt and explain what it means, indicate units.
15 10∫