Chapter 55.1 Increasing and Decreasing Functions

A function is increasing if it goes up from left to right (see figures a,b,c below), and decreasing if the function goes down from left to right (see figures d,e,f below).

Critical Numbers

1) Find the critical number (CN) and the intervals in which the function is increasing or decreasing of the following functions:a) f ( x )=x3/3−3x2/2+2 x+1. Ans: CN{1,2}; inc(-∞,1)U(2, ∞); dec(1,2)b) h ( x )=1/(x+1). Ans: no CN; dec (-∞,∞)c) I ( x )=x e−x. Ans: CN {1}; inc(-∞,1); dec(1, ∞)

d) g (t )= tt−1 . Ans: no CN; dec (-∞,1)U(1, ∞)

e) j ( x )=x−4 ln (3 x−9) Ans: CN {7}; dec(-∞,7); inc(7, ∞)f) A company that sells computers find its cost is

C ( x )=1000 x−0.2 x2 ,0≤ x≤1000 where x is the number of computers sold monthly, and its revenue can be approximate by R ( x )=0.0008 x3−2.4 x2+2400x ,0≤x ≤1000. Find when is the profit increasing.

Ans: [0,409.80)

5.2 Relative Extrema

2) Identify the x-values where the graph below has relative extrema.

Ans: Relative min at x1 and x3; Relative max at x2.

First Derivative Test:

3) Find all relative extrema of f ( x )=−x3−2x2+15 x+10. Ans: min at x=-3, max at x=5/3Find all relative extrema of g ( x )=x2/3−x5/3. Ans: no min, max at x=2/5Find the relative extrema of h ( x )=x2 ex . Ans: min at x=0, max at x=-2

4) Find the relative extrema of i (x )=2x+ln (x) . Ans: no extrema

5) Find the relative extrema of j ( x )=2x

x. Ans: min at x=1/ln(2), no max

6) Find the maximum weekly profit and the price a company should charge to realize maximum profit if the cost is given by C (q )=100+10q, and the demand is p=D (q )=50−2q. Ans: P(10)=$100; p=D (10 )=$30.

5.3 Higher Derivatives, Concavity, and Second Derivative Test

7) Find f ' ' (1 ), if f ( x )=5 x4−4 x3+3 x2−2 x+1.

Ans: f ' ' (1 )=42

8) Find the second derivative for (a) f ( x )=( x3+1 )2, (b)g ( x )=x ex, (c) h ( x )= lnxx .

Ans: f ' ' ( x )=30 x4+12x; g' ' ( x )=2ex+x ex; h' ' ( x )=2 ln ( x )−3

x3

Concavity:

9) Find the intervals where the function is concave up (CU) or concave down (CD), and find all inflection points.a) g ( x )=x3−3 x Ans: CD(-∞,0); cu(0, ∞)b) f ( x )=4 x3−x4 Ans: CD(-∞,0)U(2, ∞); CU(0,2)

Point of Diminishing Returns

This is an inflection point where adding resources (Dollars) will not increase the revenue of a business beyond that point. See figure below:

10) The revenue R ( x ) generated from the sales of certain product is related to the amount of advertising by R ( x )=(600 x2−x3)/15,000, where x and R ( x ) and in thousands of dollars. Find the point of diminishing returns. Ans:(200,1066.67)

12) Find all the relative extrema of f ( x )=−2x3+3 x2+72 x.

Ans: f ' ' (−3 )=47 min (cu); f ' ' (4 )=−47 max(cd)

5.4 Curve Sketching

Given ; ( stands for the function in the numerator and the denominator)

A) Domain of the function: Possible values of x.B) Intercepts of the function: xint solve N(x) = 0; yint= f(0) .

C) Symmetry: If the function is even (symmetric about the y-axis); if

the function is odd (symmetric about the origin);D) Asymptotes of the function:

a) Vertical (VA): solve ; Horizontal (HA): Evaluate . This also tells you the behavior of the graph far to the right and left.

E) Find the critical points (CP) by finding points where f '(x) = 0 or f '(x) is undefined. Use the first derivative test to find the intervals where f '(x) >0 (increasing) and where f '(x) <0 (decreasing). Test f '(x) in regions given by zeros, CP and VA in the domain of the function.

F) Local Maximum and Minimum Values: Find the y coordinate by evaluating f(x) at those points. Remember that not all critical points are relative extrema.

G) Concavity and Points of Inflection: Solve f "(x) = 0 for x, to find the possible inflection points. Inflection points are points where concavity changes. Not all points where f "(x) = 0 are inflection points. Find where f (x) is concave up (CU) or concave down (CD) by testing f ''(x) in regions given by

zeros, CP and VA in the domain of the function. If you evaluate at the CP, you obtain

(CU/relative min); (CD/ relative max).

H) Sketch the curve without the use of a graphics calculator. Check with a graphing calculator.

Note: Start your preliminary sketch as you find the information about the graph in parts A-G.

Examples:

1. Sketch the graph of f(x) = x3 -3x

A) Domain: all x.

B) xint x(x2-3) = 0 x = 0, x= ; yint = 0.

C) the function is odd (symmetric about the origin)

D) No asymptotes. = =E) Critical numbers at y' = 3x2 -3 = 0 (x = -1); (x = l,). Increasing when y' = 3x2 -3 > 0 on

, and decreasing when y' = 3x2 -3 < 0 on (-1,1).

F) is changes from positive to negative at x = -1 ( I(-1)=2 is a local maximum) and changes from negative to positive at x = 1 ( f(1)=-2 is a local minimum)

G) = 0 at (0,0). > 0 for x>0 (CU); < 0 for x<0 (CD). Since concavity

changes, (0,0) is an IP. |1> 0 (CU/relative min.); |-1< 0 (CD/relative max.)

H)

2. Sketch the graph of

A) Domain B) xint x = 0; yint = 0.

C) the function is even (symmetric about the y-axis)

D) VA, , HA, =1 =1

E) = 0 (x = 0,y = 0); undefined (not in the domain of f(x)); Increasing when

>0 on , and decreasing when <0 on ,F) local maximum at f(0)=0, no local minimum.

G) ; no IP. y'' = > 0 for x>0 for x <-1 or x >1 (concave up) y" = <0 for -1<x<1, (concave down).

H)

3. Sketch the graph of f ( x )= 1x2+1

a) Domain: all xb) xint none, yint =1c) Symmetry: evend) VA none; HA lim

x→±∞¿¿

e) f '( x)=−2 x /¿ ¿, CP x =0;f ' ( x )>0 for x < 0; f ' ( x )<0 for x > 0, inc(-∞,0), dec (0,∞)

f) Concavity and Inflection points: f ' ' (x )=(6x2−2)/¿¿ CU (-∞,-1/√3) U (1/√3, ∞); CD (-1/√3,1/√3), IP ( ±1/√3, ¾)

g) Graph (label all points)

h)

4. Sketch the graph of f ( x )=4 x3−x4

A) Domain: all xB) xint =0,4, yint =0C) Symmetry: noneD) VA none; HA none

E) f '( x)=12 x2−4 x3=0, CP (0,0), (3,27)f ' ( x )>0 in (-∞,3); f ' ( x )<0 in (3,∞), inc(-∞,3), dec (3,∞)

F) Concavity and Inflection points: f ' ' ( x )=24 x−12 x2=0 , x=0 , x=2 CU (0,2); CD (-∞,0) U (2, ∞); IP ( 0,0), (2, 16)

G) Graph (label all points) Scale: x=1, y=3

H)

Exercises:

a) Find the critical pointsb) Test for symmetryc) Find the intervals on which f is increasing or decreasingd) Find the intervals where f is concave up or concave down.e) Find the inflection points if any.f) Use the information in parts a-e to sketch the graph. Check with a graphing calculator.

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

Answers:

1) Inc. ; Dec. ; local max ; local min

CU ( ; CD ; IP

2) Inc. ; Dec. ; local max ;

local min ; CU ; CD ; IP

3) Inc. ; Dec. ; local max ;

local min ; CU ; CD ;

IP ,

4) Inc. ; Dec. ; local max ; local min

CU ( ; CD ; IP

5) Inc. ; Dec. ; local max ;

local min ; CU ; CD ;

IP

6) y-int :0; x-int :0; VA: x=1; HA: y =1; Dec: ; CU ; CD ; IP: none

7) y-int :-1/9; VA: ; HA: ; Inc. ; Dec: ; local max

; CU ; CD ; IP: none

8) y-int :1; x-int :1; VA: ; HA: ; Dec: ; CU ; CD ; IP: none

9) y-int :0; x-int :0; VA: None; HA: None ; Inc: ; Dec: ; local max ;

local min ; CU ; CD ; IP: ;

;

10) y-int :0; x-int :0; Inc: ; Dec: ; local max ; local min ;

CU ; CD ; IP: ;

;

Chapter 66.1 Absolute Extrema

If f is a continuous function on the interval [a, b], the absolute extrema (abs max/min) will occur either at the critical points or at the end points.

11) Find the absolute extrema of f ( x )=x2−1 on [-1, 2].

Ans: Abs min y=-1 at (0,-1); abs max y=3 at (2,3)

12) Find the absolute extrema of g ( x )=6 x2/3−35x

5/3

on [-1,8].

Ans: Abs min y=0 at (0,0); abs max y=6.6 at (-1,6.6)

If f is a continuous function on the interval (a, b), the absolute extreme values (if they exist) will occur at interior points of the interval.

13) Find the Absolute Extrema of f ( x )=x2−1 (-1, 2)

Ans: Abs min (0,-1); no abs max

6.2 Applications of Extrema (Optimization)

14) A farmer with 800 ft. of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. Use Calculus to find the largest possible total area of the four pens. Show that your area is a maximum. Show all your work.

Ans: Amax=16,000sq ft. ; A”(x)=-5 <0, relative max.

15) A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 4000 cm3. What dimensions yield the minimum surface area? Use the second derivative to check that your answer gives a minimum. Show all your work.

Ans: x=20, y=10; s”(x) =2+ 32000/x3 |x=20 >0, relative min.

16) A fence must be built to enclose a rectangular area of 20,000 ft2. Fencing material cost $2.50 per ft. for 2 sides facing north and south and $3.20 per ft. for the other 2 sides. Find the cost of the least expensive fence.

Ans: Cmin=$1600; x=160, y=125; C”(x) =256000/x3 |x=160 >0, relative min.

17) A local club is arranging a charter flight to Maui. The cost of the trip is $1600 each for 90 passengers, with a refund of $10 per passenger in excess of 90. (a) Find the number of passengers that will maximize the revenue received from the flight. (b) Find the maximum revenue.

Ans: x=35, (a)125 passengers, (b) Rmax$200,000

6.4 Implicit Differentiation

18) Given yx3 + xy = 1, find dydx

19) Given 2 = xy+x2 –y2, find dydx .

20) Given x2 y3=lny+x , find dydx .

21) Given x ey+x2=lny, find dydx .

6.5 Related Rates

22) Suppose x and y are both functions of t and x3+2 xy+ y2=1.

If x =1, and y =−2, and dxdt

=6, then find dydt.

23) A 13-ft ladder is leaning against a vertical wall. If the foot of the ladder is pulled away at a rate of ½ ft/s, how fast is the top of the ladder sliding when the lower end of the ladder is 5 ft. from the wall?

24) A company is increasing production at a rate of 2000 cases per day. All cases produced

can be sold. The daily demand function can be given by p=2000− q2

100 where q is the

number of unit produced (and sold) and p is the price in dollars. Find the rate of change of the revenue with respect to time (in days) when the daily production is 200 units.

25) Given the revenue and cost functions R=50x−0.04 x2 and C=5 x+15, where x is the daily production (and sales), find the rate of change of the profit with respect to time when 80 units are produced daily and the rate of change of production is 12 units per day.

6.6 Differentials: Linear Approximation

26) Find dyfor y=ln 2x.27) Find dyfor y=√x2+1 when x=4 and Δx=0.01.

28) Find the linear approximation of y=√x. Use the linear approximation to approximate √82.

29) Find the linearization of y=exnear x= 0, to approximate e0.01

30) Find the linearization of y=ln (x )near x=1, to approximate ln (1.05)

Marginal Analysis.

If the cost function is C (x), ∆C (x )=C ( x+1 )−C (x)≈C '(x )∆ x for ∆ x small.

Since ∆ x=1 , C ' (x) ≈C ( x+1 )−C (x) the cost of the next unit produced.

31) Let C ( x )=2 x3+300. Approximate the cost of the 101item produced, and compare with the actual cost.

32) In a precision manufacturing process, ball bearings must be made with a radius of 1.25 mm, with maximum error in the radius of ±0.025 mm. Estimate the maximum error in the volume of the ball bearing.

33) A company estimates that the revenue from the sale of certain item is given byR ( x )=12,000 ln (0.01x+1). Approximate the change in revenue from the sale of one more item, when 100 items are sold.