2.7 Notes

Write the 2 definitions of the slope of the tangent to graph of f at (a,f(a))

f(a+h)

f(a)

a+ha

h

)a(f)ha(f

h

Lim 0 h

)a(f)ha(f

h

Lim 0

ax

afxf

ax

Lim

)()(

ax

afxf

ax

Lim

)()(

h

)a(f)ha(f

h

LimMa

0

h

afhaf

h

Limaf

)()(

0)('

pointaatisthis*

)()()('

ax

afxf

ax

Limaf

Watch this video! Calculus-helpWatch this video! Calculus-help

h

afhaf

h

Limaf

)()(

0)('

pointaatisthis*

)()()('

ax

afxf

ax

Limaf

Watch this video! Calculus-helpWatch this video! Calculus-help

ExampleExample

Let Let ff((xx) = 3x) = 3x2 2 – 4x +7. Find the – 4x +7. Find the derivative of f(x) at (1, 6)derivative of f(x) at (1, 6)

The notation for this is f’(1)The notation for this is f’(1) f’(1)=2 f’(1)=2

ExampleExample

Find f’(a)Find f’(a) if if f(x)=xf(x)=x22-5x+2, -5x+2, leave your answer in terms of a. leave your answer in terms of a.

Velocity and SpeedVelocity and Speed

““VelocityVelocity is the is the derivativederivative of the position of the position function.”function.”

Also the Also the speedspeed of the particle is the of the particle is the absolute value of the velocity.absolute value of the velocity.

ExampleExample A given position function:A given position function:

ss = = ff((tt) = 1/(1 + ) = 1/(1 + tt) ) tt is seconds and is seconds and ss is meters. is meters.

Find the Find the velocityvelocity and and speedspeed after 2 after 2 seconds.seconds.

Hint:Hint: The derivative of The derivative of ff when when tt = 2 = 2

Solution (cont’d)Solution (cont’d)

Solution (cont’d)Solution (cont’d)

Thus, theThus, the velocityvelocity of the particle after 2 seconds is of the particle after 2 seconds is

f f ´(2) = – (1/9) m/s ,´(2) = – (1/9) m/s ,

and theand the speedspeed of the particle is of the particle is

||f f ´(2)| = 1/9 m/s .´(2)| = 1/9 m/s .

Estimate the derivative at the Estimate the derivative at the indicated pointsindicated points

Graph exampleGraph example

Sketch the graph of a function f for which Sketch the graph of a function f for which f(0)=2f(0)=2

f ’(0)=0 f ’(0)=0 f ’(-1)=-1f ’(-1)=-1 f ’(1)=3 f ’(1)=3 f ’(2)=1 f ’(2)=1

Let’s use a calculator!Let’s use a calculator! Let Let ff((xx) = 2) = 2xx . Estimate the value of . Estimate the value of the slope the slope

of the tangent at x=0.of the tangent at x=0. The definition of slope of a tangent givesThe definition of slope of a tangent gives

We can use a calculator to approximate the We can use a calculator to approximate the values of (2values of (2hh – 1)/ – 1)/hh . .

Use Table, then YUse Table, then Y11(small number).(small number).

0 0

0 2 1lim lim

h

h h

f h fM

h h

Solution to (a) (cont’d)Solution to (a) (cont’d)

Solution (b) from Drawing a Solution (b) from Drawing a tangent on the calculatortangent on the calculator

ReviewReview

Calculator based slope of a tangentCalculator based slope of a tangent Definition of derivative of a functionDefinition of derivative of a function Interpretation of derivative asInterpretation of derivative as

the slope of a tangentthe slope of a tangent a rate of changea rate of change

ASGN 15

p. 153 3-7odd11,13,25

Calculus-helpCalculus-help

Topics so far in Chapter 22.1 slope of a secant

average velocityaverage rate of change from table

2.2 limits from a graphlimits numerically limits piece wiselimit absolute value

2.3 limits algebraically2.4 continuity

discontinuity: removable, infinite, jumpIntermediate value theorem

2.5 limit as x goes to infinitylimit as x goes to infinityasymptotes: vertical and horizontal

2.6 Slope of a tangent lineequation of a tangent lineInstantaneous rate of change

Quiz review: Look over the

following topics, your notes, and

assignments and find a

question you would like to

clarify.