Drill: Find dy / dx
description
Transcript of Drill: Find dy / dx
Drill: Find dy/dx
1. y = sin(x2)
2. y = sec2x – tan2 x 3. y = 2x
4. y’ = 2cos(x2 )5. Let u = sec x and v = tan x
du = secxtanx and dv = sec2 x
y = u2 – v2
y’ = 2u du - 2v dvy’ = 2(sec x)(secxtanx) - 2(tanx)
(sec2x)= 0
3. y’ =2x ln 2
Fundamental Theorem of Calculus
Lesson 5.4Day 1 Homework:
p. 302/3: 1-25 ODD
Objectives
• Students will be able to– apply the Fundamental Theorem of Calculus.– understand the relationship between the derivative
and definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus, Part 1
If f is continuous on [a, b], then the function
has a derivative at every point x in [a, b], and
x
a
dttfxF
.xfdttfdxd
dxdF x
a
Example 1 Applying the Fundamental Theorem
Find
x
x
dttdxd
dttdx
d
sin
42
42 4
22
x
xsin
Note: It does not matter what a is, as long as b is ‘x.’
.xfdttfdxd
dxdF x
a
Example 2 The Fundamental Theorem with the Chain Rule
Find .cos if 2
1x
dttydxdy 2Let xu
u
tdty1
cos xdxdu 2
dxdu
dudy
dxdy
ududy cos xu
dxdy 2cos
xxdxdy 2sin 2
Example 3 The Fundamental Theorem with the Chain Rule
.cos1sin1 if 2
2
x
dttty
dxdy
xu Let
.cos1sin1
2
2
u
dttty
1dxdu
dxdu
dudy
dxdy
uu
dudy
2
2
cos1sin1
)1(cos1sin1
2
2
uu
dxdy
)(cos1)(sin1
2
2
xx
dxdy
Example 4 The Fundamental Theorem with the Chain Rule
Find . if 3
1
xx
t dtteydxdy xxu 3Let
u
t dttey1
dxdu
dudy
dxdy
13 2 xdxdu
13 23 3
xexx xx
uuedudy
)13( 2 xuedxdy u
Example 5 Variable Lower Limits of Integration
• Find dy/dx
• Need to use the rules of integrals. Remember:
• So,
• dy/dx = -3xsin(x)
• Find dy/dx5
sin3x
tdtty
a
b
b
a
dxxfdxxf )()(
x
x
tdtttdtty5
5
sin3sin3
2
2 21x
xt dt
ey
ab
a
b
dxxfdxxfdxxf00
)()()(
x
t
x
t dte
dte
y2
00 21
21
2
22
122
122
xx e
xedx
dy
xx eex
dxdy
222
22
2
Example 5 Constructing a Function with Given Derivative and Value
• Find a function y = f(x) with derivative dy/dx = tanx that satisfies the condition f(3) = 5.
• To construct a function with derivative of tanx
(always let the lower bound be the given value of x.)
• Remember that ,
• So, if x = 3, then
• Therefore, we would only need to add 5 to construct a function whose derivative is tan(x) and where f(3) = 5
x
tdty3
tan
0)( a
a
dxxf
0tan3
3
tdty
5tan)(3
x
tdtxf
Drill: Construct a function of the form that satisfies the given conditions.
x
a
Cdttfy )(
8;0;tan xyxedxdy x
3;4;cos3 xyxdxdy
x
t tdtey8
0tan
x
dtty3
4cos3
The Fundamental Theorem of Calculus, Part 2
If f is continuous on [a, b], and F is any antiderivative of f on the interval, then
This part is also called the Integral Evaluation Theorem.
.aFbFdxxfb
a
Example 6 Evaluating an Integral
Evaluate using an antiderivative.
5
3
3 564 dxxx
5
3
245
3
3 53564
xxxdxxx
3533355535 2424
536
4
1ln2 2
3
xx
Example 6 Evaluating an Integral
Evaluate using an antiderivative.
4
1
13 dxx
x
4
1
14
1
21
313 dxxxdxx
x
1ln124ln42 23
23
024ln82
4ln14
How to Find Area Analytically
To find the area between the graph of y = f(x) and the x-axis over the interval [a,b] analytically,
1) Graph the equation.2) Partition [a,b] with the zeros of f.3) Integrate f over each subinterval4) Add the absolute values of the integrals
Find the area of the region between the curve y = 4- x2, [0, 3], and the x-axis
1) Graph the equation.2) Partition [a,b] with the zeros of
f.3) Integrate f over each
subinterval4) Add the absolute values of the
integrals
• The first region, where the graph is positive, is from [0, 2]. The next region, where the graph is negative, is from [2, 3]
3
160)388(
344
2
0
32
0
2
xxdxx
37
316)912(
344
3
2
33
2
2
xxdxx
323|
37||
316|
How to find Total Area on the Calculator
• To find the area between the graph of y = f(x) and the x-axis over the interval [a, b] numerically, evaluate fnInt(|f(x)|, x, a, b) on the calculator.
• Try the previous example: fnInt(|4-x2|, x, 0, 3)• 7.66666667
Homework
• Page 303: 27-47 (ODD)