Limits Section 15-1

What You Will Learn: 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? Slope of this line Or: (x+h, f(x+h)) What will happen as “h” gets closer to zero? (x, f(x)) h

The Derivative 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 The process of finding the derivative is called differentiation. Notation for f’(x) looks like: and is read “dy, dx”

Example 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:

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

You Try 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)

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 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

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 even Homework 2: page 958, 21-27 odd, 31-41 odd