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Lesson 5.4
Q.R.: 1-10 ALL; 5.4: 1-20 (10+), 21-26 (3+), 27-40 (6+), 41-48 (4+)
Objectives
1.
2.
I feel it is important that we recap 5.1 through 5.3 before we dive in to 5.4.
5.1:
5.2:
5.3:
5.3 (Exploration Pg. 289)
Lesson 5.4 goes deeper into the connection between integration and differentiation.
The definite integral of a continuous function is a (differentiable) function of what?
THE FUNDAMENTAL THEOREM OF CALCULUS (Part 1)
This theorem tells us that every continuous function is the ____________________ of
some other function.
Additionally, every continuous function has a(n) _________________________.
What is the relationship between differentiation and integration?
Ex. 1 Find
d x
cos t dt
dx 
Ex. 2 Find
d x 1
dt
dx 0 1  t 2
CHAIN RULE REVIEW
Ex. 3 Find dy/dx if y =

x2
1
cos t dt
Ex. 4 Find dy/dx.
a) y =
5
 3t sin t dt
x
b) y =
1
dt
2 x 2  et

x2
Ex. 5 Find a function y = f(x) whose derivative dy/dx = tan x and satisfies the condition
f(3) = 5.
THE FUNDAMENTAL THEOREM OF CALCULUS (Part 2)
Ex. 6 Evaluate
3

1
( x 3  1) dx using an antiderivative.
A definite integral helps us find…
If we want, TOTAL area, we need to make sure we figure area below the x-axis
separately and assign it a positive value, then add that to all area above the x-axis.
If we see “area” we assume they want…
Ex. 7 Find the area of the region between the curve y = 4 – x2, 0 < x < 3, and the x-axis.
Ex. 8 Find the area of the region between the curve y = x cos 2x and the x-axis over the
interval -3 < x < 3.