Learning Log #6. 1.6 HW Chapter 2 Limits & Derivatives Yeah! We begin Calculus.

Learning Log #6

1.6 HW

Chapter 2Limits & Derivatives

Yeah! We begin Calculus

EXAMPLE 1

If a ball is thrown into the air with an initial velocity of 60 ft/sec, its height (in feet) after seconds is given by

a) Find the average velocity for the time period beginning when and lasting

i) .5 s.ii) .1 s.iii) .05 s.iv) .01 s.

260 16y t t t

2t

distance traveledAverage velocity =

time elapsed

260 16y t t

Example 1: Numeric

2 2

) 2 2.5

60(2.5) 16(2.5) 60(2) 16(2)

2.5 2

i from t to t

v

50 56

.5v

6

.5v

12 / secv ft

Example 1 continued

2 2

) 2 2.1

60(2.1) 16(2.1) 60(2) 16(2)

2.1 2

ii from t to t

v

55.44 56

.1v

5.6 / secv ft

Example 1 continued

2 2

) 2 2.05

60(2.05) 16(2.05) 60(2) 16(2)

2.05 2

iii from t to t

v

.24

.05v

4.8 / secv ft

Example 1 continued

2 2

) 2 2.01

60(2.01) 16(2.01) 60(2) 16(2)

2.01 2

iv from t to t

v

.0416

.01v

4.16 / secv ft

Another way to calculate average velocity …

We can calculate the velocity from time t =2 to some time slightly later t = 2+h

2120 60 16(4 4 ) 56h h h

h

2 260(2 ) 16(2 ) 60(2) 16(2)h hv

h

continued …

2120 60 64 64 16 56h h h

h

216 4h h

h

( 16 4)h h

h

continued …4 16h ) .5 4 16(.5) 12 / sec

) .1 4 16(.1) 5.6 / sec

) .05 4 16(.05) 4.8 / sec

) .01 4 16(.01) 4.16 / sec

i h ft

ii h ft

iii h ft

iv h ft

Find the instantaneous velocity at t=2.

As h approaches zero, the limiting value is -4.

4 16velocity h

(2,56)(2.5,50)

Slope of secant line = Average velocity

Slope of Tangent Line= Instantaneous velocity

Example 2

The position of a car is given by the values in the table.

a)Find the average velocity for the time period beginning when t=2 lasting… i) 3 sec ii) 2 sec iii) 1 sec

t(seconds)

0 1 2 3 4 5

s(feet) 0 10

32

70

119

178

_178 32 146) 48.6 /

5 2 3i ft s

119 32 87) 43.5 /

4 2 2ii ft s

70 32 38) 38 /

3 2 1iii ft s

Guided practice#6 Large

whiteboards

2

58 / ,

58 .83

If anarrowis shot uponthemoonwith

avelocityof m s its height is given

by h t t

) 1,2 ) 1,1.1 ) 1,1.01

d)Find theinstantaneous velocityafter1second.

a b c

Homework

Pgs.89,90 (2,5)

Due tomorrow