Calculus I

- Prof. D. Yuen

Final Exam Review version 11/22/2017

Please report any typos.

This review sheet contains summaries of new topics only.

(This review sheet does have practice problems for all topics.)

Riemann sums

• The Riemann sum associated to the definite integral dxxfb

a )( , for a uniform partition and a

choice of n and a choice of how to pick kc ’s, is

n

k

k xcf1

)( , where n

abx ,

xkaxk , and kc is in ],[ 1 kk xx . Common ways to pick kc are: left endpoints, right

endpoints, upper sum, lower sum.

Fundamental Theorem of Calculus

• If f is continuous, then )()( xfduufdx

d x

a . Combined with the chain rule, we have

)('))(()('))(()()(

)(xhxhfxgxgfduuf

dx

d xg

xh .

• The indefinite integral dxxf )( is synonymous with the antiderivative of )(xf .

• If we can find the antiderivative of )(xf , then we can evaluate the definite integral by

b

a

b

adxxfdxxf |)()( .

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(

Substitution Rule

• Usually, try to choose u the inside of a composite function.

• Compute du .

• Use the u equation and the du equation to substitute for everything in the original integral so

that the new integral has only u ’s. Use either the “pattern matching method” or the “solve

for dx method” to take care of the du . Use the method that you understand.

• Hopefully the new integral is easier to integrate. Plug back in for the original variable.

• For a definite integral, you may either change the limits of integration to the new variable, or

simply put everything back in terms of the original variable and use the original limits of

integration – but do not do both!

Areas of regions

• Decide to set up the integral using x or y as the variable based on whether you get only one

formula for the “top” curve and one formula for the “bottom” curve.

• Remember that everything must be in terms of the chosen variable.

• You must solve any equation for the other variable in terms of this chosen variable. If that

turns out to be impossible, then you must choose a different variable.

Volumes of Revolution

• Sketch the region and decide whether to do it as an x or y problem using the same

reasoning as above. x yields vertical rectangles and y yields horizontal rectangles.

• Decide whether this becomes the washer method or the shell method.

WASHER: dy

dx )radius)inner (-radius)outer (( 22 (rectangle axis of revolution)

SHELL: dy

dx height) radius)(outer (2 (rectangle || axis of revolution)

• Remember that “radius” refers to the distance as measured from the axis of revolution.

• The limits of integration come from the extreme values of the chosen variable.

Volumes by Cross-Section

• Decide on a variable axis so that cross-sections perpendicular to this axis are nice shapes.

Everything will be in terms of this variable. Label the 0 on this axis.

• Determine the area of the cross section. This is often done by writing down the area of the

cross-section shape in terms of an intermediate variable. Then the relation between the

intermediate variable and the chosen variable is obtained by looking at the global shape,

which would be the base of the solid, if there is one, of which may involve “flattening out”

the picture parallel to the variable axis. If the solid has a base as a region in the plane, then

use the equations of the bounding curves to determine this relation. Otherwise, the standard

methods for calculating this relation are: the Pythagorean theorem (if global shape is a

circle), or similar triangles (if the global shape is a triangle).

Some older topics revisited

• Sandwich theorem. For the common final exam, you might be required to write out sandwich

theorem problems fully formally. You might have to find )(xg and )(xh if not already

given, such that )()()( xhxfxg for all relevant x and say that )(lim)(lim xhLxgaxax

and hence you can conclude by the Sandwich Theorem that also Lxfax

)(lim . This also

works for limits as x . In particular, if asked to “show work”, for a Type

bounded

limit, write out the inequality based on the upper and lower bounds of the numerator:

)(

constant boundupper )(

)(

constant boundlower

xkxf

xk where )(xk , and use the

Sandwich Theorem.

• How to make a compound function continuous. The key is to compute the one-sided limits

at the crucical boundary point and set them equal to each other to make the two-sided limit

exist there.

• If given the graph of the derivative 'f and asked questions about f , then:

Questions about increasing/decreasing/max-min can be answered by looking to see where the

graph of 'f is positive, negative, zero – that is, you are pretty much given the first derivative

diagram already.

Questions about concavity can be answered by seeing where 'f is increasing (thus 0'' f ),

where 'f is decreasing (thus 0'' f ).

Additional Practice Problems The previous common final exams are already excellent sources for practice problems.

Previous exams, homework, worksheets, quizzes are also good sources;

See these other sources for extra problems not covered here.

1. Find the volumes of the following solids.

(a) The region bounded by 2xy and 22 xy rotated about the line 2x .

(b) The region bounded by xy 2 and 2 xy rotated about the y -axis.

(c) The region bounded by xy 2 , 0x and 2y rotated about the x -axis.

(d) The region bounded by yyx , 2x and 0y rotated about the line 0y .

(e) The region bounded by xy , xy 32 and 0y rotated about the line 1x

(f) The base is an isosceles right triangle with leg lengths 1. The cross-sections perpendicular

to one of the legs are semicircles.

(g) The base is the region bounded by 24 xy and 1y . Cross-sections perpendicular to

the line 1y are squares.

2. Use the Sandwich theorem to answer the following questions. Write out the Sandwich

theorem reasoning.

(a) Find x

x

x

sin72lim

.

(b) Find )cos(lim 12

0x

xx

.

(c) If 24 5)()cos(5 xxfxx for all x , can you conclude anything about )(lim0

xfx

?

What about )(lim xfx

?

3. (a) Let

25

2)(

2

xifxc

xifxcxf . What values of c makes f continuous everywhere?

(b) Let

cxifx

cxifxxg

14

1)(

3

. What values of c makes g continuous everywhere.

(a) Let

cxifcx

cxifxcxh

2)(

2

. What values of c makes h continuous everywhere.

4. Consider the graph . (a) Use implicit differentiation to find .

(b) Find the tangent line and the normal line to this graph at the point . (c) Find a

point on this graph where there is a horizontal tangent.

5. Find the derivatives of the following functions.

(a) )sec()( xxf (b) 5

)sin()(

2

x

xxg (c)

x

xduuxh

3

2

2 )1cos()(

6. Find the derivative of the following functions using only the definition.

(a) 3)( xxf (b) 32

1)(

xxf (c) xxxf 53)( 2

7. Evaluate the following limits. Show your work.

(a) x

xx

x

sin72lim

(b)

h

h

h

39lim

0

(c)

xx

x

x 7

5lim

2

3

(d)

2

2

1 )1(

23lim

x

xx

x

(e) 1

23lim

2

2

1

x

xx

x (f)

xx

xx

x sin

sin72lim

0

8. The limit in 7(b) represents the derivative )9('f for what function )(xf ?

9. (a) Suppose you are given 4

4)('

2

2

x

xxf , find where the x -values where )(xf has local

extrema. Where is )(xf increasing? Where does )(xf have inflection point(s)? Where is

)(xf concave up?

(b) Suppose you are given dtt

txg

x

0 2

2

4

4)( , find where the x -values where )(xg has local

extrema. Where is )(xg increasing? Where does )(xg have inflection point(s)? Where is

)(xg concave up?

10. Find the area of the following regions.

(a) The region bounded by 2xy and 22 xy .

(b) The region bounded by xy 2 and 2 xy .

(c) The region bounded by yyx , 2x and 0y

11. Sketch a graph having the following information:

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

1)(lim

xfx

, 1)(lim

xfx

, vertical asymptote at 1x ,

0)(' xf for 1x ; 0)(' xf for 11 x and for 1x ;.

0)('' xf for 2x and for 1x ; 0)('' xf for 12 x .

12. The acceleration at time t of an object is given by 612)( tta . Where is the object at time

1t if its initial position is 10)0( s and its initial velocity is 2)0( v ?

13. Find )(xf if xxxf 6)2sin()(' and 3)0( f .

14. Find the following integrals.

(a) dxxx )3( (b)

0 31

21 ))cos()(sin( dxxx (c) dxxx 332

(d)

0 2)cos53(

sindx

x

x (e)

1

0 3 2 17dx

x

x (f) dx

x

x )(sec2

15. Compute the Riemann sums for the integral 3

0

2 )1( dxx using 3 equal width subintervals

and (a) using right endpoints (b) using left endpoints.

16. Related rates problems: Textbook 3.7 #13, 14, 15, 17, 18, 19, 21, 22, 24, 27, 32, 35.

17. Max-min problems: Textbook 4.5 #5, 7, 8, 9, 11, 14, 16abd, 26, 32.

18. IVT and MVT problems: Textbook 2.6 #46, 47; 4.2 #15, 16.

19. Previous common finals and all your previous exams, review sheets, quizzes, worksheets,

homeworks are excellent sources of problems too.

Some Hints or Solutions to Practice Problems

1. (a) 1

1

22

3

32)2)(2(2

dxxxx

(b) 2

1

222

5

72))()2((

dyyy

(c) 1

0

222

0 21 ))2(2(

3

8))((2 dxxdyyy

(d) 1

0 15

8)2)((2

dyyyy

(e) 1

0

22 7))1()123(( dyyy

(f) 1

0

2

21

21

1

0

2

21

21 ))1((

24)( dxxdxx

(g) 2

2

22

41

15

32)1( dxx , [ 22

41 )1()( xxA ]

2. (a) Use xx

x

x

9sin725

and the sandwich theorem.

(b) Use 2122 )cos( xxxx and the sandwich theorem.

(c) Yes for 5)(lim0

xfx

. But no conclusions about )(lim xfx

because the two bounding

functions go to different limits as x .

3. (a) 21 ,2c (b) 0 ,2 ,2c (c) 0 ,3 ,3c

4. (a) xy

yxy

33

33'

2

2

(b) Tangent line is )1(

5

12 xy . Normal line is )1(52 xy . (c)

Solve 02 yx together with .

5. (a) 2/1

21)tan()sec()(' xxxxf (b) quotient rule

(c) )14cos(2)19cos(3)(' 22 xxxh

6. (a) ...33

)(' lim0

h

xhxxf

h

multiply by conjugate top and bottom

(b) ...)(' 321

3221

0lim

hxf xhx

h

combine via common denominator

(c) ...)53()(5)(3

)('22

0lim

h

xxhxhxxf

h

just expand and factor

7. (a) 0 (b) 6/1 (c) (d) (e) 2/1 (f) 2

8. xxf )(

9. (a) Local max at 2x , local min at 2x . f is increasing for 2x and for 2x .

Inflection point at 0x . Concave up for 0x .

(b) This is the same problem as (a) because '' gf .

10. (a) 1

1

22 )2( dxxx (b) 2

1

2 )2( dyyy (c) 1

0)2( dyyy

11.

12. Antidifferentiate to get 266)( 2 tttv , and 10232)( 22 tttts . Then 11)1( s .

13. 272

21 3)2cos()( xxxf

14. (a) Cxx 2

233

31 (b) 2

2

33|)sin(3)cos(2 03

121 xx (c) Substitute 33 xu

(d) Substitute xu cos53 (e) Substitute 17 2 xu (f) Substitute xu

15. (a) 830 (b) 301

16. See following pages.

17. See following pages.

18. Textbook 2.6 #46, 47; 4.2 #15, 16.

2.6#46. The key is to consider xxxf cos)( . Note that the function is continuous. Now

find an input where the output is positive and an input where the output is negative. Then

use IVT.

2.6#47. You need to apply IVT three times on three nonoverlapping intervals. So find four

inputs such that the outputs alternate positive and negative. And always note that the

function is continuous.

4.2#15. Show the existence of at least one zero by using the IVT: the function is continuous

and now find two inputs so that one output is positive and the other is negative. Show that

there cannot be more than one zero by showing that the derivative is never zero, because if

there were more than one zero, then the derivative must be zero somewhere in between by

the MVT.

4.2#16. Same technique as #15.