MAT 1221Survey of Calculus

Exam 1 Info

http://myhome.spu.edu/lauw

Expectations

(Time = 15 min.) Use equal signs correctly Use and notations correctly Pay attention to the independent

variables: Is it or ?

Tutoring Bonus Points

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Exam 1

Date and Time: 4/24 Thursday (5:30-6:50 pm)

Section 1.5, 2.1-2.5, B.1, B.2 Total Points: 80 points

Exam 1

This exam is extremely important. The second exam is on 5/15. The last

day to withdraw is 5/9. So this exam gives us the critical info for you to make a sound decision.

Calculators

Absolutely no share of calculators. Bring extra batteries, extra calculators. It is your responsibility to bring a workable calculator.

NO cell phone or PDA Your instructor/TA will not answer any

question related to calculators.

Expectations

Use equal signs Simplify your answers. Provide units. Check and Double Check your solutions. Show the “formula” steps. For word problems in B.2, show all 5

steps

Steps for Word Problems

1. Draw a diagram

2. Define the variables

3. Write down all the information in terms of the variables defined

4. Set up a relation between the variables

5. Use differentiation to find the related rate. Formally answer the question.

Major Themes:

Slope of the tangent line

Slope of the tangent line at a point on a graph can be approximated by a limiting process. (The same apply to other rate of change problems in physical sciences.)

x

Tangent Lines

To define the tangent line at x=1, we pick a point close by.

We can find the secant line of the two points

We can move the point closer and closer to x=1.

y

1

( )y f x

3

1 2

Rate of Change

y = distance dropped (ft)

x = time (s)

Find the average speed from x=2 to x=3.

2( ) 16y f x x

ft/s 801

216316

23

)2()3(

Speed

Average

22

ff

Derivative

For a function y=f(x), the derivative at x is a function f’ defined by

if it exists. (f is differentiable at x

f’(x)=The slope of the tangent line at x)

0

( ) ( )( ) lim

h

f x h f xf x

h

Limit Laws Summary

lim ( ) ( )x ah x h a

( )h x

( ) continuous at h x a ( ) not continuous at h x a

Other methods

Simplify2

1

1lim

1x

x

x

Multiply by conjugate

0

2 2limh

h

h

Differentiation Formulas

Constant Function Rule

If , then

Why?

Cxfy )( ( ) 0 0d

f x cdx

00lim0

lim

lim)()(

lim)(

00

00

hh

hh

h

h

CC

h

xfhxfxf

Constant Multiple Rule

If , then

where is a constant

( )y xk u

( )y k u x

k

( )d dku x k u x

dx dx

Power Rule

If , then

(n can be any real number)

nxxfy )( 1)( nnxxf

1n ndx nx

dx

Sum and Difference Rule

If , then)()( xvxuy )()( xvxuy

Product Rule

If ,

then

)()( xgxfy

)()()()( xgxfxgxfy

Quotient Rule

If

then

)(

)(

xg

xfy

2)(

)()()()(

xg

xgxfxgxfy

Chain Rule

( ), ( )

Therefore, ( )( )

y f u u g x

y f g x

dy dy du

dx du dx

xdx

du

udx

dydu

dyy

Extended Power Rule

dx

dunu

dx

dy

xguuy

xgy

n

n

n

1

)( ,

)(

Important Concepts

Left-hand limits and right-hand limits exists if f is continuous at a point if f is differentiable at a point if exists

lim ( )x a

f x

lim ( ) lim ( )x a x a

f x f x

lim ( ) ( )x a

f x f a

a ( )f a

Important Skills

Evaluate limits by using algebra. Finding derivatives using limits and

formula. Understand and able to perform implicit

differentiation. Solve word problems.

Remarks

Portion of points are designated for simplifying the answers.

Units are required for some answers.

Remarks

Review quiz solutions.