ESSENTIAL CALCULUS CH02 Derivatives. In this Chapter: 2.1 Derivatives and Rates of Change 2.2 The Derivative as a Function 2.3 Basic Differentiation Formulas.
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ESSENTIAL ESSENTIAL CALCULUSCALCULUS
CH02 DerivativesCH02 Derivatives
In this Chapter:
2.1 Derivatives and Rates of Change
2.2 The Derivative as a Function
2.3 Basic Differentiation Formulas
2.4 The Product and Quotient Rules
2.5 The Chain Rule
2.6 Implicit Differentiation
2.7 Related Rates
2.8 Linear Approximations and Differentials
Review
Chapter 2, 2.1, P73
Chapter 2, 2.1, P73
Chapter 2, 2.1, P73
Chapter 2, 2.1, P74
Chapter 2, 2.1, P74
Chapter 2, 2.1, P74
Chapter 2, 2.1, P74
Chapter 2, 2.1, P74
Chapter 2, 2.1, P74
Chapter 2, 2.1, P75
Chapter 2, 2.1, P75
1 DEFINITION The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope
m=line
Provided that this limit exists.
ax
afxf
)()(
X→ a
Chapter 2, 2.1, P75
Chapter 2, 2.1, P76
Chapter 2, 2.1, P76
4 DEFINITION The derivative of a function f at a number a, denoted by f’(a), is
f’(a)=lim
if this limit exists.
h
afhaf )()(
h→ 0
Chapter 2, 2.1, P77
Chapter 2, 2.1, P78
f’(a) =limax
afxf
)()(
x→ a
The tangent line to y=f(X) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a.
Chapter 2, 2.1, P78
Chapter 2, 2.1, P78
Chapter 2, 2.1, P79
Chapter 2, 2.1, P79
Chapter 2, 2.1, P79
6. Instantaneous rate of change=lim
12
12 )()(lim
xx
xfxf
x
y
∆X→0 X2→x1
The derivative f’(a) is the instantaneous rate of change of y=f(X) with respect to x when x=a.
Chapter 2, 2.1, P79
9. The graph shows the position function of a car. Use the shape of the graph to explain your answers to the following questions
(a)What was the initial velocity of the car?
(b)Was the car going faster at B or at C?
(c)Was the car slowing down or speeding up at A, B, and C?
(d)What happened between D and E?
Chapter 2, 2.1, P81
10. Shown are graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie.
(a) Describe and compare how the runners the race.
(b) At what time is the distance between the runners the greatest?
(c) At what time do they have the same velocity?Chapter 2, 2.1, P81
15. For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning.
0 g’(-2) g’(0) g’(2) g’(4)
Chapter 2, 2.1, P81
the derivative of a function f at a fixed number a:
f’(a)=limh
afhaf )()( h→ 0
Chapter 2, 2.2, P83
f’(x)=limh
xfhxf )()( h→ 0
Chapter 2, 2.2, P83
Chapter 2, 2.2, P84
Chapter 2, 2.2, P84
Chapter 2, 2.2, P84
Chapter 2, 2.2, P87
3 DEFINITION A function f is differentiable a if f’(a) exists. It is differentiable on an open interval (a,b) [ or (a,∞) or (-∞ ,a) or (- ∞, ∞)] if it is differentiable at every number in the interval.
Chapter 2, 2.2, P88
Chapter 2, 2.2, P88
Chapter 2, 2.2, P88
4 THEOREM If f is differentiable at a, then f is continuous at a .
Chapter 2, 2.2, P89
Chapter 2, 2.2, P89
Chapter 2, 2.2, P89
Chapter 2, 2.2, P89
Chapter 2, 2.2, P91
1.(a) f’(-3) (b) f’(-2) (c) f’(-1)
(d) f’(0) (e) f’(1) (f) f’(2)
(g) f’(3)
Chapter 2, 2.2, P91
2. (a) f’(0) (b) f’(1)
(c) f’’(2) (d) f’(3)
(e) f’(4) (f) f’(5)
Chapter 2, 2.2, P92
Chapter 2, 2.2, P92
Chapter 2, 2.2, P93
Chapter 2, 2.2, P93
Chapter 2, 2.2, P93
33. The figure shows the graphs of f, f’, and f”. Identify each curve, and explain your choices.
Chapter 2, 2.2, P93
34. The figure shows graphs of f, f’, f”, and f”’. Identify each curve, and explain your choices.
Chapter 2, 2.2, P93
Chapter 2, 2.2, P93
35. The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
Chapter 2, 2.2, P94
Chapter 2, 2.3, P93
FIGURE 1
The graph of f(X)=c is the line y=c, so f’(X)=0.
Chapter 2, 2.3, P95
FIGURE 2
The graph of f(x)=x is the line y=x, so f’(X)=1.
Chapter 2, 2.3, P95
DERIVATIVE OF A CONSTANT FUNCTION
0)( Cdx
d
Chapter 2, 2.3, P95
1)( xdx
d
Chapter 2, 2.3, P95
THE POWER RULE If n is a positive integer, then
1)( nnnx
dx
dx
Chapter 2, 2.3, P97
THE POWER RULE (GENERAL VERSION) If n is any real number, then
1)( nnnx
dx
dx
Chapter 2, 2.3, P97
█GEOMETRIC INTERPRETATION OF THE CONSTANT MULTIPLE RULE
Multiplying by c=2 stretches the graph vertically by a factor of 2. All the rises have been doubled but the runs stay the same. So the slopes are doubled, too.
Chapter 2, 2.3, P97
█ Using prime notation, we can write the Sum Rule as
(f+g)’=f’+g’
Chapter 2, 2.3, P97
THE CONSTANT MULTIPLE RULE If c is a constant and f is a differentiable function, then
)()]([ xfdx
dcxcf
dx
d
Chapter 2, 2.3, P97
THE SUM RULE If f and g are both differentiable, then
)()()]()([ xgdx
dxf
dx
dxgxf
dx
d
Chapter 2, 2.3, P98
THE DIFFERENCE RULE If f and g are both
differentiable, then
)()()]()([ xgdx
dxf
dx
dxgxf
dx
d
Chapter 2, 2.3, P100
Chapter 2, 2.3, P100
xxdx
dcos)(sin
xxdx
dsin)(cos
Chapter 2, 2.3, P101
Chapter 2, 2.4, P106
THE PRODUCT RULE If f and g are both
differentiable, then
)]([)()]([)]()([ xfdx
dxgxg
dx
dxgxf
dx
d
THE QUOTIENT RULE If f and g are differentiable, then
2)]([
)]([09)]([)(])(
)([
xg
xgdxd
xfxfdxd
xg
Xg
xf
dx
d
Chapter 2, 2.4, P109
xxdx
d 2sec)(tan
Chapter 2, 2.4, P110
DERIVATIVE OF TRIGONOMETRIC FUNCTIONS
xxdx
dcos)(sin xxx
dx
dcotcsc)(csc
xxdx
dsin)(cos
xxdx
d 2sec)(tan
xxxdx
dtansec)(sec
xxdx
d 2csc)(cot
Chapter 2, 2.4, P111
Chapter 2, 2.4, P112
43. If f and g are the functions whose graphs are shown, left u(x)=f(x)g(X) and v(x)=f(X)/g(x)
Chapter 2, 2.4, P112
44. Let P(x)=F(x)G(x)and Q(x)=F(x)/G(X), where F and G and the functions whose graphs are shown.
Chapter 2, 2.5, P114
THE CHAIN RULE If f and g are both differentiable and F =f 。 g is the composite function defined by F(x)=f(g(x)), then F is differentiable and F’ is given by the product
F’(x)=f’(g(x))‧g’(x)
In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then
dx
du
du
dy
dx
dy
dx
dyF (g(x) = f’ (g(x)) ‧ g’(x)
outer evaluated derivative evaluated derivative
function at inner of outer at inner of inner
function function function function
Chapter 2, 2.5, P115
Chapter 2, 2.5, P116
4. THE POWER RULE COMBINED WITH CHAIN RULE If n is any real number and u=g(x) is differentiable, then
Alternatively,
dx
dunuu
dx
d nn 1)(
)(')]([)]([ 12 x‧xgnXgdx
d n
Chapter 2, 2.5, P120
49. A table of values for f, g, f’’, and g’ is given
(a)If h(x)=f(g(x)), find h’(1)
(b)If H(x)=g(f(x)), find H’(1).
Chapter 2, 2.5, P120
51. IF f and g are the functions whose graphs are shown, let u(x)=f(g(x)), v(x)=g(f(X)), and w(x)=g(g(x)). Find each derivative, if it exists. If it dose not exist, explain why.
(a) u’(1) (b) v’(1) (c)w’(1)
52. If f is the function whose graphs is shown, let h(x)=f(f(x)) and g(x)=f(x2).Use the graph of f to estimate the value of each derivative.
(a) h’(2) (b)g’(2)
Chapter 2, 2.5, P120
Chapter 2, 2.7, P129
█WARNING A common error is to substitute the given numerical information (for quantities that vary with time) too early. This should be done only after the differentiation.
Chapter 2, 2.7, P129
Steps in solving related rates problems:1. Read the problem carefully.
2. Draw a diagram if possible.
3. Introduce notation. Assign symbols to all quantities that are functions of time.
4. Express the given information and the required rate in terms of derivatives.
5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3).
6. Use the Chain Rule to differentiate both sides of the equation with respect to t.
7. Substitute the given information into the resulting equation and solve for the unknown rate.
Chapter 2, 2.8, P133
Chapter 2, 2.8, P133
f(x) ~ f(a)+f”(a)(x-a)~
Is called the linear approximation or tangent line approximation of f at a.
Chapter 2, 2.8, P133
L(x)=f(a)+f’(a)(x-a)
The linear function whose graph is this tangent line, that is ,
is called the linearization of f at a.
Chapter 2, 2.8, P135
dy=f’(x)dx
The differential dy is then defined in terms of dx by the equation.
So dy is a dependent variable; it depends on the values of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f, then the numerical value of dy is determined.
Chapter 2, 2.8, P136
r
dr
r
drr
V
dV
V
V3
344
3
2
relative error
Chapter 2, Review, P139
1. For the function f whose graph is shown, arrange the following numbers in increasing order:
Chapter 2, Review, P139
7. The figure shows the graphs of f, f’, and f”. Identify each curve, and explain your choices.
Chapter 2, Review, P140
50. If f and g are the functions whose graphs are shown, let P(x)=f(x)g(x), Q(x)=f(x)/g(x), and C(x)=f(g(x)). Find (a) P’(2), (b) Q’(2), and (c)C’(2).
Chapter 2, Review, P141
61. The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.
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