1
Chapter 4 Applications of Differentiation
Section 4.1 Maximum and Minimum Values
1. Identify the relative (local) maximum (maxima) or minimum (minima) and the absolute (global) maximum (maxima) or minimum (minima) for
each function.
Relative extrema only occur in open intervals (not at endpoints!) and Absolute extrema can also occur at endpoints and when the derivative is zero
or undefined.
(a) The point (1, 3) is not on the graph.
(b) x --> infinity on the right side
(c) The endpoints are not on the graph.
(d) Point (3, 6) not on graph.
(e) Graph from [  , 2 ] .
(f) Points (0,2) and (5,5) not on graph.
To begin the process of finding extrema, we first need to learn about critical numbers.
A critical number is where the derivative is undefined or zero.
Examples of critical numbers:
(a) The derivative is zero
(b) The derivative is undefined
5
4
f '(2)=0
4
3
f '(0)=und
2
2
1
-5
-2
2
4
5
6
-2
-1
-2
-4
In example (a) x
undefined there.
 2 is a critical number since at that point the derivative is zero. In example (b) x  0 is a critical number since the derivative is
Critical numbers are very important, since that’s where maxima and minima might occur.
2. Find the critical numbers of the following functions.
(a)
2
3
y  ( x  3)  1
(b)
y
x2
x2 1
(c) y 
1
sin 2 x
2
2
Absolute(global) extrema
The absolute maxima and minima occur when you are given a function which is restricted to a specific interval of
3. Find the absolute max and mins of the following:
(a)
y  x 2e x on the interval [-3, 1].
f ( x) 
(b)
1
5
cos 2 x  sin x , [0, ].
2
6
(c)
x.
y  x3  3x 2 ,  1  x  2 .
4.2 The Mean Value Theorem MVT
Rolle’s Theorem
i) f is continuous on
a  x  b.
ii) f is differentiable on a  x  b.
iii) f  a   f  b 
Conditions:
Then there is a number
1. Verify the hypothesis of Rolle’s Theorem and find
(a)
f  x   2  2x  2x2 ,
c such that f   c   0 on a  x  b.
c.
2, 1
(b) f  x   3sin 2 x,
  
  2 , 2 
Mean Value Theorem
Let f by a function that satisfies the following hypothesis:
a  x  b.
ii) f is differentiable on a  x  b.
i) f is continuous on
Then there is a number
c such that f   c  
2. Portion of the graph of
f b   f  a 
ba
.
f  x   x3  3x2  2, [1, 1] has been graphed below.
(a) Find the slope of the secant line whose x-coordinates are x = -1 and x = 1. Sketch the secant line.
(b) Use the mean value theorem to find c.
The MVT helps you find the value at which the function will have a tangent line with a slope equal to that of the secant line.
3
3. Find the value(s) of c guaranteed by the mean value theorem for each function listed below. Sketch each function and the secant line as well as
the tangent line(s) on the coordinate plane provided.
(b) f ( x)  2 x  1 ,
(a) f ( x)  x( x  2)( x  2) [-2, -1].
4. Consider the graph of the function
y  f ( x) 
0  x  3.
x
. Is it possible to apply the MVT to locate a value of c on the interval [0, 3].
( x  2) 2
Explain.
4.3 How Derivatives affect the shape of a graph
Analyzing the graph of a function
f (x) >0 implies that the y-values of the derivative are above the x- axis and f (x) <0 implies that the y-values of the f (x) derivative
are below the x-axis.
6
f'x = 3x2+-6x
4
2
-2
2
4
-2
-4
-6
-8
f x = x3-3x2
1. The graph of f (x ) is given. Determine the intervals for which the function f (x ) is increasing or decreasing.
(a)
(b)
2. Given the graph of f(x) Determine intervals where f (x ) > 0 or f (x ) < 0. Sketch the graph of f (x ) on the same x – y plane.
(i)
Is the derivative increasing or decreasing? Explain (Recall when I used a ruler to demonstrate tangent lines to the curve)
a)
(b)
6
f x =
1
4
x
2
-10
10
-2
-4
3. Find the set of values of
4. Given
x for which the function defined as y  5  4 x 2  2 x 3 is decreasing as a function of x .
f ( x)  x 2  4 x  3 .
(a) Determine the interval’s where the function
f ( x)  x 2  4 x  3 is increasing and decreasing. Justify your answer.
(b) Determine the maximum or minimum value of the function.
5. Determine the interval(s) where the function
6. Let f  x  
f ( x)  3x 4  4 x 3  12 x 2  2 is increasing and decreasing.
4
, then find the intervals where the function increases and decreases.
x 1
2
7. Find the value of k, where f ( x)  k x  2kx  11, and f (1)  5.
4
5
8. Use the information on the following table to sketch the graph of f(x).
x
0
1
2
3
0
-2
-4
0
f x
 
f (x)
f (x) >0
x0
f (x) <0
0 x2
0
f (x) >0
x2
Finding relative extrema
Relative maxima and minima are also known as local maxima and local minima. A specific interval is not given.
9. Find the relative max and minimums of the following functions.
(a)
10. Let
f ( x) 
x4
 x3  2 x 2  12 x  2
4
f ( x)  x x  4
(b)
x(t )  a  bet where “a” and “b” are constants, and contains the points P(0,1) and Q  ln 2, 1 .
(a) Find the value of a and b.
(b) A secant line is drawn at the points where t  0 and t  ln 8 . For what value of “ t ”, will the secant line be tangent to the curve “ x
t  ”?
p
have a minimum when x  5 ?
x3
To answer the question, evaluate the derivative at x  5 and set that equal to zero.
11. for what value of p will x  1 
12. Find the absolute maxima and minima of the function f ( x)  x cos x, 0  x 
5
. also identify the relative minimum and maximum if
2
any.
13. Suppose that f (1)  4 and that f ( x)  10  x, how large can f (3) possible be?
14. Given the function
f ( x)  x 2 e x where 6  x  1.
(a) For what values of x is f increasing? Decreasing?
(b) Determine all extrema.
15. Find the x-coordinate of the relative maxima and minima for ln
 2  cos x 
0  x   . Determine where the function is increasing and
decreasing on the specified interval.
16. A twice differentiable function has the following properties: Use for parts (a) – (c) below.
I.
+
f ( x) 

-2
0
6
6
II.
+
+
+
f ( x) 

-8
0
2
(a) Is f ( 6)  f (3) ? True or False
(b) Is f ( 1)  f ( 3) ? True or False.
(c) Is f (1)  f (10) ? True or False.
Concavity
A continuous differentiable function is concave up whenever f ( x ) is increasing.
A continuous differentiable function is concave down whenever f ( x ) is decreasing.
Note: A function f ( x ) increases when f ( x ) > 0 and decreases when f ( x ) < 0, therefore f ( x ) must increase when f ( x) >0 and
f ( x) must decrease when f ( x) < 0.
A function changes concavity at a point of inflection. Possible points of inflection occur when f ( x) is zero or undefined. To be a
point of inflection the second derivative must change sign.
17. The function is y 
15
f x =
1 4
x  2x2 .
12
15
 
1
12
15
x4-2x2
10
10
f'x =
5
-5
x3
3
10
f'' x = x2+-4
+-4x
5
5
5
-5
5
-5
5
-5
-5
-5
-10
-10
-10
-15
-15
-15
-20
-20
-20
First find the critical numbers of the first and second derivatives. (questions continue on the next page)
What do the zeros of the second derivative correspond to in
f ( x)?
f  x?
What happens when the first derivative is increasing and decreasing?
What happens when the second derivative decreases and or increases?
What do the max or mins of the second derivative correspond to on f ( x ) ?
18. On the following graph, approximate the intervals where the function is concave up and
down. Also approximate the points of inflection. Draw tangents at
f x = x+22xx-1 
4
3
x  2,  1.5,  1, 0, and 1.
2
1
-2
2
-1
-2
7
f (t )  t 2 (t  3), t  0 . Determine the times for which the particle is moving to the
left and to the right. Is the particle accelerating or decelerating on the interval [0, 2]?
19. The velocity of a particle is described by the equation
20. The following graphs are the graphs of the second derivative of a twice differentiable function.
(a) What does the graph tell us about
f  x ?
I
(b) What does the graph tell us about
II.
21. The following graph is the graph of
f  x ?
III.
f   x  . Determine where f(x) is concave up and concave down. What else can you state about f(x)?
(a)
(b)
22. Determine the points of inflection and the intervals in which the graph is concave up and down.
x3 x 4

(a) y 
3
2
23.. Determine the points of inflection of
(b)
y  ( x  4)e x
1 5
x  x4 .
5
Test the critical numbers of the first derivative for relative extrema by using the second derivative test.
24 Determine the relative extrema of:
(a) y 
1 3 5 2
x  x  50 x
3
2
(b)
f ( x)  xe x
25. The function f ( x)  (2  x) g ( x) where g is twice differentiable on the interval
Determine if x
 2 yields a maximum or a minimum.
0  x  4. Let g ( x)  0 when x >1, and g  2  0 .
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26. The derivative of a continuous and differentiable function f(x) for x > 0 has 1 
(a) If f (1)  1 , what is the equation of the tangent line to f at
x  1?
1
, x  0 as its derivative.
x
(b) Use the equation of the tangent line to f found in part (a) to approximate f (1.1).
(c) At x 
1
will the tangent line to f be above the curve or below the curve? Explain your answer.
2
27. Differentiate:
(a)
f ( x)  1  ln(sin x)
(b) f ( x)  csc 2 x  cot 2 x  sec 2 x
28. Sketch the graph of the polynomial function f(x). Use the information given on the table.
x  4
x
f ( x)
f ( x)
x  4
34
4  x  0
x0
x0
30
0  x 1
x0
x 1
28.5
x0
x 1
x0
40
30
20
10
-6
-4
-2
2
4
6
-10
-20
-30
-40
dy
for:
dx
t
(a) x  e and y  cot t
29. Find
30. The function
(b)
csc( x  y)  x  y 2
(c)
x  xy  y 2  6, when y  2
f ( x)  30 x5  5 x3  45 x has a relative maximum value at x  ?
31. The derivative of
f ( x)  (3x 2  2 x3 ) has a minimum value of ___?
32. The radius of a set of concentric circles is decreasing at a constant rate of 2 feet per second. How fast is the area of the circles changing 6
seconds later when the radius has reached a length of 6 feet?
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4.4 L’Hôpitals Rule
L’Hôpitals Rule is used to evaluate limits that produce indeterminate forms such as the ones listed below:
1.
0
0
2.


3.   
4.
0 
5. 0

0
6. 1
7.
0
L’Hôpitals Rule
 a, b  containing c , except possibly at c itself. Assume that
x in  a, b  except possibly at c itself. If the limit of f  x  / g  x  as x approaches c produces the indeterminate
f  x
f  x
0




, or
, or
, or
form , or
, then lim
.
 lim
x c g  x 
x c g   x 
0




Let f and g be functions that are differentiable on an open interval
g   x   0 for all
1. Evaluate each limit:
sin 3 x
=
x 0
x
(a) lim
e2 x  e2
(b) lim

x 1
x 1
ex
(c) lim 3 
x  x
x99
(d) lim 3 x 
x  e
(e) lim
x 0
tan x  5 x
=
x3
4.5 and 4.6 Curve Sketching
Sketch the graph of the following functions without using a calculator. Use symmetry, x and y intercepts, asymptotes, the first and second derivative.
Determine the domain and range of each function.
(a)
y  x3  3x 2 .
(b)
y  x4  2x2
(c)
y  x2 , 0  x   .
(d) h( x ) 
x
x 1
2
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4.7 Optimization
1. The sum of two numbers is 500. Find the largest product of the numbers.
2. What is the shortest distance from the point
1, 5 to the curve given by the following:
1
y  x2 .
2
3. Find the area of the largest rectangle that can be inscribed in a semicircle of radius 4.
4. Find the volume of the largest cylinder that can be inscribed in a sphere of radius 8 m.
5. A rectangle is inscribed in the first quadrant by the coordinate axes and a line which passes through the points (7, 0) and (0, 5). Find the area of
such rectangle.
6. Maximize the area of the rectangle which is bounded by the x axis and inscribed by the curve y  2 cos x,


2
x

2
.
7. Jerry the farmer wants to enclose his pigs, horses, and cows, within adjacent corrals. He has 2000 feet of fencing. If he has to divide the corral
into three stables, what is the maximum area he can enclose with the amount of fencing given.
8. If y  4  2 x , what is the smallest value of xy ?
9. The dimensions of a sheet of metal are 40 in by 30 in. Squares of side x inches are cut from each corner, and what remains is folded to make a
box without a lid.
(a) Write an expression for the volume V of the box.
(b) State the range of possible values of x.
(c) Find the x – coordinate for which the volume is of greatest size.
(d) Find the maximum value of V.
Section 4.9 Antiderivatives – note that 4.8 isn’t included in this packet
Integrate:
n
 x dx 
x n 1
 C , n  1. This is the power rule for integrals.
n 1
 kdx  kx  C, k 
. This is the constant rule for integrals.
 f ( x)dx   g ( x)dx  F ( x)  G( x)  C . Addition and Subtraction rule for integrals.
 0dx  C . The integral of zero is a constant.
1. Antidifferentiate:
(a) 
 x  2  dx
(d) 
 2e
x

 3sin x dx
1  tan 2 y
dy
(g) 
tan 2 y



(e)   2 x 
(h)


x 1  x 2 dx
(b) 
1
sec 2 x  e
2
5
dx
1  x2
(c) 

 dx

1  x  x2
3
x
dx
(f)  1  4x  dx
2
(i)

50
1 x
2
dx
(j)

b 

 ax  2 dx
x 

11
2. Find the particular solution of the differential equation which satisfies the given initial conditions.
1
  0, and f  2   40
3
(a) f ( x )  36 x, f  
3. Find
(b)
f   x   et  sin t ,
f   0  2,
f    1
f  x  given that f   x   12x  e x
y
2
 x 2 , then find  ydx REWRITE
5. A function y  f ( x) passes through the point (1, 6). Given that its derivative
x
dy
3
8
 4  5 , x  0 , show that the point (2, 5) is on the curve f ( x ).
is
dx x
x
4. If
6. If y  f ( x)  sin x , what is the value of




d
 ydx
dx


d
d
d
  cos x  C   s in x
 ydx =
 sin xdx 
dx
dx
dx
7. A ball is launched from the roof of a building 40 feet high with an initial velocity of 60 feet per second. When will the ball hit the ground?
Therefore the path of the particle is
equation.
8. Find the velocity function when
s(t )  16t 2  60t  40 and to find the time at which the ball hits the ground, find the zeros of the
v  2   3 and a(t ) 
2t
.
t3
9. A ball is thrown vertically upward from the ground with an initial velocity of 96 ft/s. How high will the ball go? When will it hit the ground?
To find out how high it will go, find the when velocity is zero, set s(t) = 0 to find when it hits the ground.
10. An object has a constant acceleration of 42 ft/ s2 and an initial velocity of -18ft/s and an initial position of 3 feet. Find the equation describing
the motion of the object.
11. A ball is dropped from an initial height of 300ft. Its velocity after t seconds is - 32 ft/s.
(a) How fast is the ball dropping after 4 seconds?
(b) Determine the position function
(c) How far has the ball dropped after 4 seconds?