Calculus I

- Prof. D. Yuen

Exam 3 Review version 11/4/2017

Check website for any posted typos and updates

Extrema and Concavity

• A critical point of a function is a number in the domain where the derivative is 0 or DNE.

• Local extrema can only occur at critical points or end points (but not every critical point is a

local extremum).

• To find the absolute extrema of a continuous function on a closed finite interval, one method

is to compare the function values at all the critical points and end points.

• To find the absolute extrema of a function on any interval, find the critical points and use the

first derivative test to understand the graph.

• f is increasing when 0'f and f is decreasing when 0'f .

• [First derivative test] A critical point is a local max when 'f changes from to , and is a

local min when 'f changes from to .

• f is concave up when 0'' f and f is concave down when 0'' f .

• f has an inflection point where ''f changes sign.

• When investigating increasing/decreasing and concavity using first derivative and second

derivative line diagrams (sign charts), remember to include locations of vertical asymptotes

as well as critical points and possible inflections, respectively.

Horizontal and Vertical Asymptotes

• f has a horizontal asymptote (H.A.) at the horizontal line Ly if Lxfx

)(lim (asymptote

is to the right) or Lxfx

)(lim (asymptote is to the left).

• f has a vertical asymptote (V.A.) at the vertical line ax if

or )(lim xfax

and/or

or )(lim xfax

.

• Candidates for vertical asymptotes usually come from where the denominator of the function

is zero but the numerator is not zero.

Sketching a graph

• Use facts about domain, intercepts, horizontal asymptotes, vertical asymptotes, local

extrema, increasing/decreasing, inflection points, concavity. Put it all together and think. It

usually helps to draw the line diagrams for increasing/decreasing and for concavity

synchronously above where you will sketch the graph. It is usually better to pencil in any

asymptotic parts of the graph first.

• You may be required to sketch a graph given the function or given derivative information or

other information.

Max-Min word problems

• Read the problem carefully, draw a diagram and assign variables. Do not assign variables to

constant quantities.

• Set up an objective function (to be maximized or minimized), which might at first involve

more than one input variable. Also, write down any relations (constraints) among the input

variables.

• If the objective function is in more than one input variable, then you must use the constraints

to eliminate down to one input variable.

• Note the domain of the objective function in this one input variable. You usually find the

domain by considering for what inputs does the function make sense. This often involves

looking at any constraints, if there are any.

• Take a derivative and find the critical points in the domain.

• Find the absolute maximum or minimum. If the domain is a closed finite interval, then you

could use the method of comparing the function values at the critical points and end points.

Or, in any case, you could use the first derivative to understand the graph (for example, you

could be lucky and there is only one critical point and the signs of the derivative would tell

you exactly where the desired absolute extremum is).

Newton's Method

• To numerically find a solution to 0)( xf , the recursion is )('

)(1

n

n

nnxf

xfxx .

Antiderivatives

• Some antiderivative rules are:

Cxn

dxx nn

1

1

1 for 1x gdxbfdxadxbgaf )(

Cbaxa

dxbax )cos(1

)sin( Cbaxa

dxbax )sin(1

)cos(

Cbaxa

dxbax )tan(1

)(sec2 Cbaxa

dxbaxbax )sec(1

)tan()sec(

Practice Problems Your homework, worksheets, quizzes are also good sources.

1. Find the absolute extrema of:

(a) 12)( 23 xxxf on ]2 ,1[ . (b) xxxg )cos(2)( on ] ,0[2 .

2. Find the critical points of:

(a) 7

)(20

13

x

xxf (b) xxxg 2)( 3/1

3. Find all local extrema of:

(a) 51834)( 23 xxxxf (b) )1()( 5/1 xxxg

4. Find all inflection points of:

(a) 133)( 234 xxxxxf (b) 3 433)( xxxg

5. Sketch a graph having the following information:

0)4( f , 0)2(' f , 0)4('' f ,

1)(lim

xfx

,

)(lim xfx

, vertical asymptote at 0x ,

0)(' xf for 0x and for 2x ; 0)(' xf for 20 x .

0)('' xf for 0x and for 40 x ; 0)('' xf for 4x .

6. Sketch a graph having the following information:

1)0( f , 0)2( f , 0)1(' f , 0)3('' f ,

)(lim xfx

, 2)(lim

xfx

, vertical asymptote at 0x ,

0)(' xf for 10 x ; 0)(' xf for 0x and for 1x .

0)('' xf for 3x ; 0)('' xf for 0x and for 30 x .

7. A farmer has 400 ft of fence and wishes to make 4 identical side by side pig rectangular pens

in a 1 by 4 formation. What dimensions will maximize the total area?

8. A farmer wishes to make 4 identical side by side rectangular pig pens in a 1 by 4 formation

with each pig pen having area 160 square feet. What dimensions will minimize the length of

fence used?

9. A box with no top is to be 2 feet wide. If the volume is to be 36 cubic feet, what should the

other two dimensions be so as the minimize the material?

10. We want to make a box where the length is twice the width. The bottom of the box costs 1.5

as much as the rest of the material. If the box is to have volume 90 cubic feet, what

dimensions will minimize the cost?

11. Write the recursion in Newton's method for numerically solving 013 xx . Starting with

10 x , find 2x . Leave it as a fraction.

12. Find the following antiderivatives:

(a) dxxxx )7/2412( 33 (b) dxxx ))2cos(10)sin(10(21 (c) dxxx )2)(1(

13. Find )(xf if 5)4cos()3sin()(' xxxf and 1)0( f .

14. The acceleration at time t of an object is given by tta 620)( . Where is the object at time

2t if its initial position is 10)0( s and its initial velocity is 5)0( v ?

Solutions to Practice Problems 1. (a) Max is 1 at 2,0x and Min is 2 at 1x . (Critical points at ),0

34x

(b) Max is 6

3 at 6x and Min is

2 at

2x . (Critical points at

6x )

2. (a) Critical points at 2020 13,13,0 x (Derivative is 220

1232

)7(

917

x

xx.)

(b) Critical points at 2161

2161 ,,0 x (Derivative is 23/2

31 x .)

3. (a) Local max at 1x , Local min at 23x .

(b) Local min at 61x .

4. (a) Inflection Points at 1,21x . (b) Inflection Points at 0,

23x .

5.

6. Other solutions possible, such as the vertical asymptote going to -infinity as x approaches 0

from the right.

7. Each pen should be 25 ft by 40 ft (the sides that are "shared" have length 40 ft). The total

area is 4000 square feet.

8. Each pen should be 10 ft by 16 ft (the sides that are "shared" have length 16 ft). The total

length of fence is 160 feet.

9. Dimensions are 2 ft wide by 6 ft long by 3 ft tall.

10. Dimensions are 3 ft wide by 6 ft long by 5 ft tall.

11. The recursion is 13

12

3

1

n

nn

nnx

xxxx . 2/31 x , 23/312 x .

12. (a) Cxxxx 7128 22/34

41 (b) Cxx )2sin(5)cos(20

21 (c) Cxxx 22

233

31

13. 34

41

31 5)4sin()3cos()( xxxxf

14. 10510)( 32 tttts and position at time 2 is 52)2( s at time 2t .