Transcript Lesson 6-2

Lesson 5-3b
Fundamental Theorem of
Calculus
Quiz
• Homework Problem:
x + 7sec2x) dx
(
3e
∫
= 3ex + 7tan x + c
• Reading questions: Fill in the squares below
█
b
∫█af(x) dx
b – F(█)
a
= F(█)
If F’(x) = f(x)
Objectives
• Understand both forms of the Fundamental
Theorem of Calculus
• Understand the relation between integration
and differentiation
Vocabulary
• Definite Integral – is a number, not a function
Differentiation vs Definite Integration
∆x
∆x
∆y
∆y
∆x
∆x
∆y
∆y
Area
under the
curve
Area of
rectangle
a
b
a
∆y
f(b) – f(a)
slope = ------ = -------------∆x
b–a
b
a
a
b
area = ∆y ∆x
= f(b) (b-a)
∆y
slope ≈ ------∆x
∆y
lim ---- = mt = f’(a)
∆x→0 ∆x
b
area ≈ ∆y ∆x
b
lim ∆y ∆x = area =
∆x→0
∫a f(x) dx
Fundamental Theorem of Calculus, Part 1
If f is continuous on [a,b], then the function g defined by
x
g(x) =
∫af(t) dt
a≤x≤b
is continuous on [a,b], differentiable on (a,b) and g’(x) = f(x)
Your book has a proof of this if you are interested.
We can see g’(x) = f(x) by using derivatives and FTC part 2.
Fundamental Theorem of Calculus
Combining parts 1 and 2 of the FTC, we get
x
g(x) =
∫af(t) dt
= F(x) – F(a)
F(x) is a function of x and F(a) is just some constant value.
Now when we do the following:
d
---dx
x
g(x) =
∫af(t) dt
g’(x) =
g’(x) = f(x)
= F(x) – F(a) =
= F’(x) – 0
F’(x)
= f(x)
Example Problems
Find the derivative of each of the following:
x
1)
∫ t² dt
1
2)
∫
2
x
t3/2
------------ dt
t² + 17
t=x
⅓t³ |
[⅓x³ ] – [⅓1³ ]
We can’t do what
we did in #1!
d(⅓x³ – ⅓) / dx = x²
Use the pattern:
t=1
See any pattern?
x3/2
-----------x² + 17
Example Problems with TI-89
Find the derivative of each of the following:
x
1)
∫ t² dt
= x²
1
Can we use our calculator here? YES!!
Hit F3 select derivatives; hit F3 select integration; type
in function (t²), integrate with respect to (t), lower
limit of integration (1), upper limit of integration (x);
close ). Type , and differentiate with respect to x and
close ). Should look like this:
d( F3 ∫(t^2,t,1,x),x)
Example Problems cont
Find the derivative of each of the following:
π/4
4
3)
∫ tan²(t) cot (t) dt
x
4)
∫ t tan (t) dt
x
We can’t do what
we did in #1!
We can’t do what
we did in #1!
Use the pattern:
Use the pattern:
- tan²(x) cot(x)
-x tan (x)
Negative because of
location of x in integral
Negative because of
location of x in integral
Example Problems cont
Find the derivative of each of the following:
x²
2x
5)
∫ 5t sin (t) dt
1
6)
∫ √2 +
sin (t) dt
2
We can’t do what
we did in #1!
We can’t do what
we did in #1!
Use the pattern:
Use the pattern:
10x sin(2x) (2)
Remember Chain Rule
√2 + sin (x²)
(2x)
Remember Chain Rule
Example Problems cont
Find the derivative of each of the following:
x³
7)
∫ √1 + t
4
dt
x
8)
∫
cos x
t² dt
sin x
We can’t do what
we did in #1!
We can’t do what
we did in #1!
Use the pattern:
Use the pattern:
√1 + (x³)4 (3x2) - √1 + x4
cos²x (-sinx) - sin²x(cosx)
Remember F(b) – F(a)
and chain Rule
Remember F(b) – F(a)
and Chain Rule
Summary & Homework
• Summary:
– Definite Integrals are a number
– Evaluated at endpoints of integration
– Derivative of the integral returns what we
started with (with Chain Rule)
• Homework:
– Day One: pg 402-404: 19, 22, 27, 28,
– Day Two: pg 402-404: 3, 7, 9, 61 (see
appendix E)
Second Set Example Problems
Find the derivative of each of the following:
x²
1)
∫ sin³(t) dt
1
We can’t do what
we did in #1!
Use the pattern:
sin³(x²) (2x)
Remember Chain Rule
Second Set Example Problems
Find the derivative of each of the following:
5
2)
∫ (1/t²)dt
x
We can’t do what
we did in #1!
Use the pattern:
- (1/x²)
Negative because of
location of x in integral
Second Set Example Problems
Find the derivative of each of the following:
x²
3)
∫ √1 + t
4
dt
x
We can’t do what
we did in #1!
Use the pattern:
√1 + (x²)4 (2x) - √1 + x4
Remember F(b) – F(a)
and chain Rule
Last Example Problem
Consider F (x) =
Find F  (x).
∫
0
x
t–3
----------- dt
t² + 7
for -∞ < x < ∞
x–3
F’(x) = ----------x² + 7
For what number x does F attain its minimum value?
______ Justify your answer.
When F’(x) = 0
so x = 3
First Derivative Test: - 0 + therefore a min
How many inflection points does the graph of F have?
_____ Justify your answer.
When F’’(x) = 0
so x = -1, x = 7