Mean Value Theorem for Derivatives4.2

If you drive 100 miles north

…in 2 hours…

What was your average velocity for the trip?

50 miles/hour

Does this mean that you were going 50 miles/hour the whole time?

No. Were you at any time during the trip going 50 mi/hr?

Absolutely. There is no way that you couldn’t have been.

100 miles

Remember Mr. Murphy’s plunge from the diving platform?

sec05.3

)0()5.3(

feetss

vavg

sec05.3

1960

feet

vavg

sec5.3

196 feetvavg

= 56 feet/sec

…is an equation that you know finds…

…the slope of the line through the initial and final points.

secm from time 0 to time 3.5…Mr Murphy’s average velocity from 0 to 3.5 seconds

s(t)

= H

eigh

t off

of

the

grou

nd(i

n fe

et)

t = Time in seconds

216196)( tts

Is there ever a time during Mr Murphy’s fall that his instantaneous velocity is also –56 feet/sec?

ttsvinst 32)(.

= 1.75 seconds

Absolutely. We just need to find out

where s´(t) = –56 feet/sec

This means that the slope of the secant line through the initial and final points…

…is parallel to the slope of the tangent line through the point t = 1.75 seconds

s(t)

= H

eigh

t off

of

the

grou

nd(i

n fe

et)

t = Time in seconds

sec/5632.

feettvinst

2sec/32

sec/56

feet

feett

216196)( tts

It is also the point at which Mr. Murphy’s instantaneous velocity is equal to his average velocity

216196)( tts

If the function f (x) is continuous over [a,b] and

differentiable over (a,b), then at some point

between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

Average Velocity

Instantaneous Velocity

y

x0

A

B

a b

Slope of chord:

f b f a

b a

Slope of tangent:

f c

y f x

Tangent parallel to chord.

c

If the function f (x) is continuous over [a,b] and

differentiable over (a,b), then at some point

between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

Differentiable implies that the function is also continuous.

If the function f (x) is continuous over [a,b] and

differentiable over (a,b), then at some point

between a and b:

f b f af c

b a

Mean Value Theorem for DerivativesMean Value Theorem for Derivatives

Differentiable implies that the function is also continuous.

The Mean Value Theorem only applies over a closed interval.

If the function f (x) is continuous over [a,b] and

differentiable over (a,b), then at some point

between a and b:

f b f af c

b a

Mean Value Theorem for DerivativesMean Value Theorem for Derivatives

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.