1. The Derivative in Graphing and Applications
By: Tair Akhmejanov

2. Section 5.1:Analysis of Functions
Increasing
Decreasing
Concavity
Inflection Points

3. To describe functions we need a few definitions
A function is increasing on an interval if f(x1)<f(x2) when x1<x2
A function is decreasing on an interval if f(x1)>f(x2) when x1>x2
A function is constant on an interval if f(x1)=f(x2) for all x1 and x2
Thus, a function is increasing on an interval where it has a positive slope, decreasing on an interval where it has a negative slopes, and constant on an interval where its graph has zero slope.
Let f be a function that is continuous on a closed interval [a,b] and differentiable on the open interval (a,b).
If for every value of x in (a,b), then f is increasing on [a,b]
If for every value of x in (a,b), then f is decreasing on [a,b]
If for every value of x in (a,b), then f is constant on [a,b]

4. Concavity
If f is differentiable on an open interval I, then f is concave up on I if is increasing on I, and f is concave down on I if is decreasing on I.
Let f be differentiable twice on an open interval I.
If on I, then f is concave up on I.
If on I, then f is concave down on I.
If f is continuous on an open interval containing a value xo, and if f changes the direction of its concavity at the point (xo,f(xo)), then we say that f has a n inflection point at xo, and we call the point (xo,f(xo)) on the graph of f an inflection point of f.

5. Try It
State when is increasing, decreasing, concave up, concave down, and all of the inflection points. + -
f(x) is decreasing on (- ,-3] U [-1, )
f(x) is increasing on [-3,-1]
f(x) is concave down on (-2, )
f(x) is concave up on ( , -2)
f(x) has an inflection point at (-2,-2)

6. Try It
State when is increasing, decreasing, concave up, concave down, and all of the inflection points.
f(x) is increasing on (- ,-8)U(-8, )
f(x) is never decreasing
f(x) is concave up on (- , -8)
f(x) is concave down on (-8, )
f(x) has no inflection points

7. Section 5.2 Relative Extrema First Derivative Test
Second Derivative Test

8. Suppose that f is a function defined on an open interval containing the number xo. If f has a relative extremum at x=xo then either or f is not differentiable at xo.
First Derivative Test
If on an open interval extending left from x and on an open interval extending right from xo, then f has a relative maximum at xo.
If on an open interval extending left from x and on an open interval extending right from xo, then f has a relative minimum at xo.
If has the same sign on an open interval extending left from xo and on an open interval extending right from xo, then f does not have a relative extremum at xo.
Second Derivative Test
If and > 0, then f has a relative minimum at xo.
If and < 0, then f has a relative maximum at xo.
If and = 0, then the test is inconclusive; that is, f may have a relative maximum, a relative minimum, or neither at xo.

9. Try It Find the relative extremum of
By second derivative test f(x) has a relative maximum at (0,0), and relative minimums at
(2,-4) and (-2,-4)
By second derivative test f(x) has a relative minimum at
and a relative maximum at

10. Section 5.4
Rectilinear motion

11. Rectilinear Motion
If s(t) is the position function of a particle moving on a coordinate line, then the instantaneous velocity of the particle at time t is defined by:
Instantaneous speed at time t:
If s(t) is the position function of a particle moving on a coordinate line, then the instantaneous acceleration of the particle at time t is defined by:
*Note*: A particle in rectilinear motion is speeding up when its velocity and acceleration have the same sign and slowing down when they have opposite signs.

12. Reminders about particles
When the velocity is negative, the particle is moving to the left.
When the velocity is positive, the particle is moving to the right.
When the velocity and acceleration of the particle have the same signs, the particle’s speed is increasing.
When the velocity and acceleration of the particle have opposite signs, the particle’s speed is decreasing (or slowing down).
When the velocity is zero and the acceleration is not zero, the particle is momentarily stopped and changing direction.

13. Try It If the position function of a particle is , t>0
find when the particle is changing directions.
If the position function of a particle is , t>0 when is the particle speeding up?
Velocity is never 0 so particle never changes direction
(4, )

14. Section 5.5
Absolute Maxima and Minima

15. Extreme-Value Theorem
A function f is said to have an absolute maximum on an interval I at xo if f(xo) is the largest value of f on I; f(xo) f(x) for all x in the domain of f that are in I.
A function f is said to have an absolute minimum on an interval I at xo if f(xo) is the smallest value of f on I; f(xo) f(x) for all x in the domain of f that are in I.
Extreme-Value Theorem
If a function f is continuous on a finite closed interval [a,b], then f has both an absolute maximum and an absolute minimum on [a,b].
If f has an absolute extremum on an open interval (a,b), then it must occur at a critical number of f.
*Note*: Calculate all critical values of f and calculate values at endpoints.
Largest value is the absolute maximum and smallest value is the absolute minimum.
Suppose f is continuous and has exactly one relative extremum on an interval I at xo.
If f has a relative minimum at xo, then f(xo) is the absolute minimum of f on I.
If f has a relative maximum at xo, then f(xo) is the absolute maximum of f on I.

16. Try It
At what value of x does f(x) have an absolute minimum given that
At what point does f(x) have an absolute minimum and at what point does f(x) have an absolute maximum given that
on the interval 4≤x≤9
By the second derivative test f(x) has an absolute minimum at
There is an absolute minimum at (5.24,10.47)

17. Section 5.6
Applied Maximum and Minimum Problems

18. Suggestions for Applied Maximum and Minimum Problems
Draw a figure and label the quantities relevant to the problem.
Find a formula for the quantity to be maximized or minimized
Using the conditions stated in the problem to eliminate variables, express the quantity to be maximized or minimized as a function of one variable.
Find the interval of possible values for this variable from the physical restrictions in the problem.
If applicable, use the techniques of the preceding section to obtain the maximum or minimum.
Note: Profit = Revenue - Cost
P(x) = R(x) - C(x)

19. Try It Find the point on closest to point (4,0)
Use the distance formula.
Plug in for y.
Let L(x)=D^2 because the minimum value of D^2 will occur at the same value of x as the minimum value of D.

20. Try It
Given a rectangular sheet of cardboard 18 inches by 24 inches a box is made by cutting a square with sides x inches from each corner and bending the resulting sides up. Find the value of x that maximizes the volume of the box.
Width: 18-2x
Length: 24-2x
Depth: x
Volume=x(18-2x)(24-2x)
Eliminate x=10.6 because it does not work in the context of the problem.
Volume of the box maximized at x=3.4

21. Section 5.7
Newton’s Method
, n=1,2,3…

22. Newton’s Method
Newton’s Method is used to find the point at which the graph of a function intersects the x-axis.
Usually you are given a starting value, allowed to use a calculator to pick a starting value, or required to make an educated guess about the root to pick a good starting value.

23. Try It
Given a function , x>0 find its root using Newton’s Method. Start at x1=2.1
Given a function , x>0 find its root using Newton’s Method (you may use a calculator to get your starting value for x)
The more times this method is repeated by plugging in the new value for x, the more accurate the approximation of the root will be.
In this case the root is x=2. X=
A question will usually tell you to run the test a certain amount of times.

24. Section 5.8
Mean-Value Theorem
Rolle’s Theorem

25. The Mean Value Theorem (for derivatives)
Let f be differentiable on (a,b) and continuous on [a,b]. Then there is at least one number c in (a,b) such that:
Another way of putting it:
Given a function f
there is a point c in an interval (a,b) where the slope of the tangent line equals the slope of the secant line of the endpoints of the interval.

26. Rolle’s Theorem
Let f be differentiable on (a,b) and continuous on [a,b]. If f(a)=f(b)=0, then there is at least one number c in (a,b) such that f’(c)=0.
Another way of putting it: The Rolle’s theorem is just an instance of the Mean Value Theorem when the slope is zero, and therefore the tangent line is a horizontal tangent.

27. Try It Find the values of c that satisfy the Mean Value Theorem
on the interval [-4,4]

28. Try It Find values of c that satisfy the Mean Value Theorem for
on the interval [-1,2]
Find the values of c that satisfy Rolle’s Theorem for
on the interval [0,1]
No Solution
C=1/2

29. QUIZ 1) Find the values of c that satisfy Rolle’s Theorem for
on the interval [-1,1]
2) Find the absolute maximum and minimum values of on the interval [-3,3]
3) A rectangle is inscribed in a semicircle, that has a radius of 4, with one side on the semicircle’s diameter. What is the largest area this rectangle can have? A) B) C) D)
min.= B) min.= C) min= D) min.=-.385
max= max= max= max=.385
A) B) C) D) 9

30. 4) A rectangular field, bounded on one side by a building, is to be fenced in on the other three sides. If 3,000 feet of fence is to be used, find the dimensions of the largest field that can be fenced in.
5) Find the values of c that satisfy the Mean Value Theorem
for on the interval [0,4]
6) Find the point at the relative maximum and the point at the relative minimum of
A) 750 ft. by 1,000 ft B)500 ft. by 1,500 ft C)350 ft. by 2,300 ft D)750 ft. by 1,500 ft.
A) B) C) D)
A)max.:(4,0) B)no relative maximum C)max:(3,8) D)max:(5,24)
min:(3,-27) min:(3,-27) no relative minimum min:(-1,-14)

31. 7) A poster is to contain 100 square inches of picture surrounded by a 4-inch margin at the top and bottom and a 2-inch margin on each side. Find the overall dimensions that will minimize the total area of the poster.
8) Given the position function , find when the particle is speeding up.
9) If the position function of a particle is <t<4 , find when the particle is changing direction.
A) 5in. by 5in. B) 5in. By 3.5in. C) in. by in. D) in. by in.
A) 2<t<7/ B) 2<t<7/2 and t> C)2<t<3 and t> D)4>t>7
A)t= , B)t= / C)t= , / D)4

32. 10) Given a function state when it is concave up, concave down, and the points of inflection.
Cocave up: (-3,0)U(2, )
Concave down: (- ,-3)U(0,2)
Inflection point(s): (-3,189); (0,0); (2, -16)
B) Concave up: ( , )
Concave down: never
Inflection point(s): none
D) Concave up: (- ,0)U(2, )
Concave down: (0,2)
Inflection point(s): (2,-16)
C) Concave up: (- ,0)U(2, )
Concave down: (0,2)
Inflection point(s): (2,0)

33. Credits This Presentation: Tair Akhmejanov Fact Checker: Alex Bishop
Text Book: Anton, Bivens, Davis
Princeton Review: David S. Kahn
Barron’s: Shirley O. Hockett, David Bock
PowerPoint: Microsoft Corp.

34. Bibliography of Pictures