ESSENTIAL CALCULUS CH02 Derivatives

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ESSENTIAL CALCULUSCH02 Derivatives
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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

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1 DEFINITION The tangent line to the curve yf(x)
at the point P(a, f(a)) is the line through P
with slope mline
Provided that this limit exists.
X? a
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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? 0
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f(a) lim
x? a
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The tangent line to yf(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.
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6. Instantaneous rate of changelim
?X?0
X2?x1
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The derivative f(a) is the instantaneous rate of
change of yf(X) with respect to x when xa.
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• 9. The graph shows the position function of a
  car. Use the shape of the graph to explain your
  answers to the following questions
• What was the initial velocity of the car?
• Was the car going faster at B or at C?
• Was the car slowing down or speeding up at A, B,
  and C?
• What happened between D and E?

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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?
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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)
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the derivative of a function f at a fixed number
a f(a)lim
h? 0
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f(x)lim
h? 0
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3 DEFINITION A function f is differentiable a if
f(a) exists. It is differentiable on an open
interval (a,b) or (a,8) or (-8 ,a) or (- 8, 8)
if it is differentiable at every number in the
interval.
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4 THEOREM If f is differentiable at a, then f is
continuous at a .
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• (a) f(-3) (b) f(-2) (c) f(-1)
• (d) f(0) (e) f(1) (f) f(2)
• (g) f(3)

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2. (a) f(0) (b) f(1) (c) f(2)
(d) f(3) (e) f(4) (f) f(5)
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33. The figure shows the graphs of f, f, and f.
Identify each curve, and explain your choices.
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34. The figure shows graphs of f, f, f, and
f. Identify each curve, and explain your
choices.
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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.
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FIGURE 1 The graph of f(X)c is the line yc, so
f(X)0.
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FIGURE 2 The graph of f(x)x is the line yx, so
f(X)1.
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DERIVATIVE OF A CONSTANT FUNCTION
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THE POWER RULE If n is a positive integer, then
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THE POWER RULE (GENERAL VERSION) If n is any real
number, then
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GEOMETRIC INTERPRETATION OF THE CONSTANT
MULTIPLE RULE
Multiplying by c2 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.
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Using prime notation, we can write the Sum Rule
as (fg)fg
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THE CONSTANT MULTIPLE RULE If c is a constant and
f is a differentiable function, then
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THE SUM RULE If f and g are both differentiable,
then
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THE DIFFERENCE RULE If f and g are both
differentiable, then
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THE PRODUCT RULE If f and g are both
differentiable, then
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THE QUOTIENT RULE If f and g are differentiable,
then
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DERIVATIVE OF TRIGONOMETRIC FUNCTIONS
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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)
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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.
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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
yf(u) and ug(x) are both differentiable
functions, then

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F (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
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4. THE POWER RULE COMBINED WITH CHAIN RULE If n
is any real number and ug(x) is differentiable,
then Alternatively,
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49. A table of values for f, g, f, and g is
given

1. If h(x)f(g(x)), find h(1)
2. If H(x)g(f(x)), find H(1).

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• 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.
• u(1) (b) v(1) (c)w(1)

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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)
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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.
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• Steps in solving related rates problems
• Read the problem carefully.
• Draw a diagram if possible.
• Introduce notation. Assign symbols to all
  quantities that are functions of time.
• Express the given information and the required
  rate in terms of derivatives.
• 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).
• Use the Chain Rule to differentiate both sides of
  the equation with respect to t.
• Substitute the given information into the
  resulting equation and solve for the unknown rate.

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f(x) f(a)f(a)(x-a)

Is called the linear approximation or tangent
line approximation of f at a.
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The linear function whose graph is this tangent
line, that is , is called the linearization
of f at a.
L(x)f(a)f(a)(x-a)
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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.
dyf(x)dx
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relative error
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1. For the function f whose graph is shown,
arrange the following numbers in increasing order
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7. The figure shows the graphs of f, f, and f.
Identify each curve, and explain your choices.
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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).
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61. The graph of f is shown. State, with reasons,
the numbers at which f is not differentiable.
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