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Some Math 135 review problems

The following problems are based heavily on problems written by Professor Stephen Green-

eld for his Math 135 class in spring 2005. His willingness to let these problems be used

this semester is gratefully acknowledged. The problems are grouped by topic.

Denition of derivative

1. Write the denition of derivative as a limit and use this denition to nd the derivative

of f(x)=x

2

x.

2. Write the denition of derivative as a limit and use this denition to nd the derivative

of f(x)=

1

3x+4

.

3. Write the denition of derivative as a limit and use this denition to nd the derivative

of f(x)=

p

12x.

Computations of limits

1. Evaluate the limits exactly. Give brief evidence supporting your answers which is not

based on a calculator graph or on calculator computations.

a) lim

x!1

2x

2

5

3x

2

+1

b) lim

x!4

x4

p

x2

c) lim

x!5

x

2

3x10

x5

d) lim

x!+1

2x

3

x+3

x

3

+2

e) lim

x!2

x

2

4

j2xj

f) lim

x!2

+

x

2

4

j2xj

2. Find the equations of all vertical and horizontal asymptotes of f(x)=

p

x

2

+4

x3

.

3. Find the equations of all vertical and horizontal asymptotes of f(x) =

3+5e

2x

7e

2x

.

Computations with exp and log should be simplied as much as possible; approximations

are not acceptable.

Computations of derivatives

1. Find

dy

dx

.

a) y=

2x

2

+5

5x

3

+1

b) y=e

p

x

2

1

c) xy

3

= cos(7x+5y) d) xe

y

= sin(xy)

e) y=xln(3x+5) f) 2x

3

+5x

2

y+y

3

=2 g) y= ln(x

4

+3x+1) h) y=

3x

3

1

3x

4

+1

i) y=x

5

tan(3x) j) y=(4x+3)

p

x

3

+7

2. Suppose f(x) is a dierentiable function satisfying f

0

(1) = 3 and f

0

4

=2, If

g(x)=f(tanx), nd g

0

4

.

3. An equation of the tangent line to y=f(x) when x=1isknown to be 2x+y+3=0.

Find f(1) and f

0

(1).

4. Suppose Uis a dierentiable function with U(8) = 5 and U

0

(8) = 3, and that V(x) =

U(x

3

). What are V(2) and V

0

(2)? Use this information to write an equation of the tangent

line to y=V(x) when x=2.

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5. A function is dened implicitly by the equation x

2

y

2

5xy +x+y= 12. Find y

0

in

terms of xand y. Find an equation for the line tangent to the graph of this function at

the point (2;1).

Log/exp etc.

1. Find the range of f(x)=e

2x

+e

3x

.

Continuity & dierentiability

1. Here f(x)=

x+3 if x2

1

2

x

2

+Aif 2<x

where Ais a constant to be determined. Find Aso

that f(x)iscontinuous for all values of x. Sketch a graph of y=f(x) using that value of

Afor 4x2. Is f(x) dierentiable at x=2 using that value of A?

2. Here f(x)=

(

Ax

2

1 if x<1

x+Bif 1x1

2 if 1 <x

where Aand Bare constants to be determined.

Find numbers Aand Bso that f(x) is continuous for all values of x. Sketch a graph of

y=f(x) for 3x3.

3. In this problem f(x) =

8

<

:

1+x

2

if x<2

A+Bx if 2x<1

x

2

if x1

. Find Aand Bso that f(x) is

continuous at all points. Sketch a graph of y=f(x) for 3x3. For which values of

xis f(x)not dierentiable?

4. In the graph of y=f(x) to the right,

identify with many point which is a

relative minimum; Many point which

is a relative maximum; Cany point

which is a critical point; Iany point

which is an inection point; NC any

point at which f(x)isnot continuous;

and ND any point at which f(x)isnot

dierentiable. Some points may have

more than one label.

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