Limits

Section 15-1

WHAT YOU WILL LEARN:

1. How to find the derivatives and antiderivatives of polynomial functions.

Derivatives and Antiderivatives

• Imagine you wanted to solve the following problem:

- Suppose a ball is dropped from the upper observation deck of the CN Tower, 450 meters above the ground.

a. What is the velocity of the ball after 5 seconds?

b. How fast is the ball traveling when it hits the ground?

We’ll come back to this.

Remember Our Old Friend Slope?

x-10 -5 5 10

y

1

2

3

4

5

xhx

xfhxfm

)(

)()(

h

xfhxf )()(

(x+h, f(x+h))

(x, f(x))

Slope of this line

Or:

h

What will happen as “h” gets closer to zero?

The Derivative

h

xfhxfh

)()(lim

0

• So…the formal definition of a derivative:

• This gives us the slope of a line tangent to a point on the curve. Another way to say this would be the rate of change of the function at that particular point.

Differentiation

dx

dy

• The process of finding the derivative is called differentiation.

• Notation for f’(x) looks like: and is read “dy, dx”

Example

dx

dy• Find an expression for the slope of the tangent line to

the graph of y = x2 – 4x + 2 at any point. In other words, find .

• Step 1: find f(x+h):

• Step 2: find:

• Step 3: find:

h

xfhxf )()(

h

xfhxfh

)()(lim

0

Example (continued)

42 xdx

dy

• Using the derivative of the function we just found, find the slopes of the tangent line when x = 0 and x = 3.

You Try

dx

dy• Find an expression for the slope of the tangent line to

the graph of y = 2x2 – 3x + 4 at any point (find ).

• Find the slopes of the tangent lines when x = -1 and x = 5.

Rules for Finding Derivatives of Polynomials

• Constant Rule: The derivative of a constant function is 0. If f(x) = c then f’(x) = 0

• Power Rule: If f(x) = xn, where n is a rational number, then f’(x) = nxn-1.

• Constant Multiple of a Power Rule:

If f(x) = cxn, where c is a constant and n is a rational number, then f’(x) = cnxn-1.

• Sum and Difference Rule: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x).

Examples• Find the derivative of the following:

1. f(x) = x6

2. f(x) = x2 – 4x + 2

3. f(x) = 2x4 – 7x3 + 12x2 – 8x – 10

More Examples• Find the derivative:

4. f(x) = x3(x2 + 5)

5. f(x) = (x2 + 4)2

You Try• Find the derivative of each function.

1. f(x) = x5 4. f(x) = x6 – x5 – x4

2. f(x) = x3 + 2x 5. f(x)=(x + 1)(x2 – 2)

3. f(x) = 2x5 – x + 5

Antiderivatives• We can work “backwards” from a derivative back

to a function. Very helpful for moving from velocity or rate of change back to the original function.

• Example. Find the antideriviative of the function f’(x) = 2x.

• We know it is x2 but what about x2 + 1, x2 + 2, x2 + 3…

Rules

Cxn

xF n

1

1

1)(

• Power Rule: If f(x) = xn, where n is a rational number other than -1, the antiderivative is:

• Constant Multiple of a Power Rule: If f(x) = kxn, where n is a rational number other than -1 and k is a constant, the antiderivative is:

• Sum and Difference Rule: If the antiderivatives of f(x) and g(x) are F(x) and G(x) respectively, then the antiderivative of

f(x) + or – g(x) is

1

1

1)(

nxn

kxF

)()( xGxF

Examples• Find the antiderivative of each function.

1. f(x) = 3x7

2. f(x) = 4x2 – 7x + 5

3. f(x) = x(x2 + 2)

You Try• Find the antiderivative of each function.

1. f(x) = 32x3

2. f(x) = 35x6 + 12x2 – 6x + 12

3. f(x) = x2(x2 + x + 3)

Word Problems• Page 958, #46

You Try• Page 959, #49

Summary• Derivative of functions = rate of change of the

function. Measures how fast a function changes.

• Antiderivative of functions = if you are given an rate of change, you can work your way back to the original function (less c). If you are given a point from the original function, you can even “recover” a value for c.

Homework

Homework 1: Page 958, 14-18 evenHomework 2: page 958, 21-27 odd, 31-41 odd