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