MATH 151 Engineering Mathematics Iroquesol/Math_151_Week_4.pdf · MATH 151 Engineering Mathematics...
Transcript of MATH 151 Engineering Mathematics Iroquesol/Math_151_Week_4.pdf · MATH 151 Engineering Mathematics...
Chapter 3. Derivatives
MATH 151 Engineering Mathematics ISpring 2017, WEEK 4
JoungDong Kim
02/14/2017
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Definition. DerivativeThe Derivative of a function f at a number a, denoted by f ′(a), is
f ′(a) = limh→0
f (a + h)− f (a)
hor
f ′(a) = limx→a
f (x)− f (a)
x − a
if this limit exists.
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Ex10) Find the derivative of the following functions at the numbera given. (use definition)
a) f (x) = x2 − 8x + 9, a = −2
f ′(a) = limh→0f (a + h)− f (a)
h
f ′(−2) = limh→0f (−2 + h)− f (−2)
h
f ′(−2) = limh→0
((−2 + h)2 − 8(−2 + h) + 9
)−((−2)2 − 8(−2) + 9
)h
f ′(−2) = limh→04− 4h + h2 + 16− 8h + 9− 29
h
f ′(−2) = limh→0h2 − 12h
h= limh→0
h(h − 12)
h= limh→0(h − 12) = −12
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
b) f (x) =x
x + 1, a = 2
f ′(a) = limh→0f (a + h)− f (a)
h
f ′(2) = limh→0f (2 + h)− f (2)
h
f ′(2) = limh→0
((2+h)
(2+h)+1
)−(
(2)(2)+1
)h
f ′(2) = limh→0
(2+h3+h
)−(23
)h
f ′(2) = limh→0
(2+h)3−2(3+h)3(3+h)
h
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
f ′(2) = limh→0(6 + 3h)− (6 + 2h)
3h(3 + h)
f ′(2) = limh→0h
3h(3 + h)
f ′(2) = limh→01
3(3 + h)=
1
9
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Interpretations of the Derivatives
Geometrically the tangent line to y = f (x) at (a, f (a)) is the linethrough (a, f (a)) whose slope is equal to f ′(a), the derivative of fat a.
1 The slope of the tangent line to the graph of f (x) atx = a.
2 The instantaneous rate of change of f (x) at x = a.
3 The instantaneous velocity at x = a.
All use
limh→0
f (a + h)− f (a)
h= f ′(a)
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Ex11) Recall the surface area of a sphere is given by A = 4πr2.Find the average rate of change of the area from r = 1 to r = 2.Find the instantaneous rate of change of the area at r = 1.
Ave. rate of change =A(rfinal )− A(rinitial )
rfinal − rinitial
=A(2)− A(1)
2− 1=
4π(2)2 − 4π(1)2
1= 16π − 6π = 12π
A′(a) = limh→0A(a + h)− A(a)
h
A′(1) = limh→0A(1 + h)− A(1)
h
A′(1) = limh→04π(1 + h)2 − 4π(1)2
h
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
A′(1) = limh→04π((1 + h)2 − 1
)h
A′(1) = limh→04π(1 + 2h + h2)− 4π
h
A′(1) = limh→08πh + 4πh2
h=
h(8π + 4πh)
h
A′(1) = limh→0 (8π + 4πh) = 8π
or
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
A′(a) = limx→aA(x)− A(a)
x − a
A′(1) = limx→1A(x)− A(1)
x − 1
A′(1) = limx→14π(x)2 − 4π(1)2
x − 1
A′(1) = limx→14π((x)2 − 1
)x − 1
A′(1) = limx→14π(x − 1)(x + 1)
x − 1
A′(1) = limx→1 4π(x + 1) = 8π
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Definition. Differentiable A function f is differentiable at a iff ′(a) exists.
Theorem. If f is differentiable at a, then f is continuous at a.
How can a function FAIL to be differentiable
1 If the graph of a function f has a “corner” or “kink” in it,then the graph of f has no tangent at that point and f is notdifferentiable there.
2 If f is not continuous at a, then f is not differentiable at a.
3 The curve has a vertical tangent line when x = a, f is notdifferentiable at a.
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Ex12) The graph of f is given. State, with reasons, the numbers atwhich f is not differentiable.
1.- The function is not differentiable at x = −2 because thefunction f has a discontinuity at that point.
2.- The function is not differentiable at x = −1 because thefunction f has a corner at that point.
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
3.- The function is not differentiable at x = 4 because the functionf has a discontinuity at that point.
4.- The function is not differentiable at x = 8 because the functionf has a corner at that point.
5.- The function is not differentiable at x = 11.2 because thefunction f has a vertical tangent line at that point.
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Ex13) Where is f (x) = |x2 − 4| not differentiable.
Solution
-4 -3 -2 -1 0 1 2 3 4
x
-4
-2
0
2
4
6
8
10
12
f
Function f(x)= x2
- 4
-4 -3 -2 -1 0 1 2 3 4
x
0
2
4
6
8
10
12
f
Function f(x)= | x2
- 4 |
Ex14) Where is f (x) =|x + 1|x + 1
not differentiable.
SolutionJoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
The function f is defined by
f (x) =|x + 1|x + 1
=
{−1 x < −1
1 x > −1
and its graph is given by
-6 -4 -2 0 2 4
-5
-4
-3
-2
-1
0
1
2
3
4
5Function f(x)= abs(x + 1)/(x+1)
Therefore the function f is not differentiable at x = −1 because fis not continuous there.
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Ex15) Given the graph of f (x) below, sketch the graph of thederivative.
a)
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
The graph of the derivative is
-6 -4 -2 0 2 4
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Function df/dx
2/3
-5/9
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
b)
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
The graph of the derivative is
-5 0 5
-15
-10
-5
0
5
10
15Function df/dx
eb
c
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Definition. Derivative of functionThe derivative of f is defined as
f ′(x) = limh→0
f (x + h)− f (x)
h
Ex16) Find the derivative of the following functions as well as thedomain of the derivative.
a) f (x) = x2 − 8x + 9
f ′(x) = limh→0f (x + h)− f (x)
h
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
f ′(x) = limh→0
((x + h)2 − 8(x + h) + 9
)−(x2 − 8x + 9
)h
f ′(x) = limh→0
((x2 + 2xh + h2)− (8x + 8h) + 9
)−(x2 − 8x + 9
)h
f ′(x) = limh→0
(x2 + 2xh + h2 − 8x − 8h + 9
)−(x2 − 8x + 9
)h
f ′(x) = limh→0
(2xh + h2 − 8h
)h
= limh→0h(2x + h − 8h)
h
f ′(x) = 2x − 8
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
b) f (x) =3x + 1
x − 2
f ′(x) = limh→0f (x + h)− f (x)
h
f ′(x) = limh→0
3(x+h)+1(x+h)−2 −
3x+1x−2
h
f ′(x) = limh→0
[3(x+h)+1][x−2]−[3x+1][(x+h)−2][(x+h)−2][x−2]
h
f ′(x) = limh→0
[3x+3h+1][x−2]−[3x+1][x+h−2][x+h−2][x−2]
h
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
f ′(x) = limh→0
[3x2+3hx+x−6x−6h−2]−[3x2+3xh−6x+x+h−2][x+h−2][x−2]
h
f ′(x) = limh→0
−7h[x+h−2][x−2]
h
f ′(x) = limh→0−7h
h[x + h − 2][x − 2]=
−7
(x − 2)2
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
c) f (x) =4√x
f ′(x) = limh→0f (x + h)− f (x)
h
f ′(x) = limh→0
4√x+h− 4√
x
h
f ′(x) = limh→0
4√
x−4√
x+h√x+h√
x
h
f ′(x) = limh→04[√x −√x + h]
h√x + h
√x
[√x +√x + h
√x +√x + h
]
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
f ′(x) = limh→04x − 4(x + h)
h√x + h
√x [√x +√x + h]
f ′(x) = limh→0−4h
h√x + h
√x [√x +√x + h]
f ′(x) = limh→0−4√
x + h√x [√x +√x + h]
f ′(x) =−4√
x√x [√x +√x ]
f ′(x) =−4
x [2√x ]
=−2
x√x
=−2
x3/2
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
Ex17) The limit below represent the derivative of some functionf (x) at some number a. Identify f (x) and a for each limit.
a) limh→0
(2 + h)5 − 32
h
= limh→0
(2 + h)5 − 25
h
= limh→0
f (2 + h)− f (2)
h; f (x) = x5
= limh→0
f (a + h)− f (a)
h; f (x) = x5; a = 2
= limh→0
f (a + h)− f (a)
h= f ′(a); f (x) = x5; a = 2
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4
Chapter 3. Derivatives Section 3.1 Derivatives
Derivatives
b) limx→3π
cos x + 1
x − 3π
= limx→3π
cos x − cos(3π)
x − 3π
= limx→3π
f (x)− f (3π)
x − 3π; f (x) = cos(x)
= limx→3π
f (x)− f (a)
x − a; f (x) = cos(x); a = 3π
= limx→3π
f (x)− f (a)
x − a= f ′(a); f (x) = cos(x); a = 3π
JoungDong Kim MATH 151 Engineering Mathematics I Spring 2017, WEEK 4