Warm Up – NO CALCULATORLet f(x) = x2 – 2x.1) Determine the average rate of change of f(x)

over the interval [-1, 4].

2) Determine the value of(Check your answer using your calculator)

4

1f (x)dx

Mean Value Theorem for Integrals

Average Value

2nd Fundamental Theorem of Calculus

Mean Value Theorem for Integrals

If f is continuous on [a,b] then there is a certain point (c, f(c)) between a and b so if you draw a rectangle whose length is the interval [a,b] and

whose height is f(c), the area of the rectangle will be exactly the area beneath the function on [a,b].

a bc

In other words…

*If f is continuous on [a,b], then there exists a number c in the open interval (a,b) such that ...

( ) ( )( )b

af x dx f c b a

Area under the curve from a to b

Area of the rectangle formed

Example 1: Find the value of f(c) guaranteed by

MVT for integration for the function f(x) = x3 – 4x2 + 3x + 4 on [1,4]

Explain the relationship of this value to the graph of f(x)?

Example 2Find the value of c guaranteed by MVT for integrals on the interval [1,9] for

1f(x) x2

Example 3Find the value of c guaranteed by MVT for integrals on the interval [0,π/3] for

f(x) = sinx

The f(c) value you found in the examples is called the average value of f. Solving for f(c) gives the formula for average value.

1( ) ( )

b

a

Average Value f c f x dxb a

Example 4: Find the average value of f(x) = 3x2 – 2x on the interval [1,4] and all values of x in the interval for which the function equals its average value.

Example 5: Calculator activeFind the average value of on the interval [1,9] and all values of x in the interval for which the function equals its average value.

)1x2(x)x(g

Taking the derivative of a definite integral whose lower bound is a number and whose upper bound contains a variable.

f (x)

a

d g(y)dydx

The long way…

4x

2

d 3t dtdx

)x('Fthen,dtt3)x(FIfx

2

The 2nd Fundamental Theorem of Calculus:If f(x) is continuous and differentiable,

f (x)

a

d g(y)dy g(f(x))f ' (x)dx

Here’s what you REALLY do…

x

2

d 3t dtdx

4x

2

d 3t dtdx

Your turn…

2

3

x2

x

0

12

t

d1) (y 6y)dydx

d2) (cos )ddx

d3) 2x 3dxdt

If 2x

2t 3

1F x dt , then F' 3

)x('Fthen,dt1t2)x(FIf5x3

x

Let f be defined on the closed interval [-5,5]. The graph of f consisting of two line segments and two semicircles, is shown above.

f

1

5

( ) f x dx5

5

( )

f x dx

Find g(2)

1

( ) ( )x

g x f t dt

f

Find g’(2)

Find g”(2)

Let g be the function given by

1

( )x

f t dt g(x)=

f

On what intervals, if any, is g increasing?

Find the x-coordinate of each point of inflection of the graph g on the open interval (-5,5). Justify your answer.

1

( )x

f t dt g(x)=

f

Find the average rate of change of g on the interval [-5,5].