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Limits Section 15-1

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What You Will Learn: How to find the derivatives and antiderivatives of polynomial functions.

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**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.

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**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

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**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.

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

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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:

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

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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.

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**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).

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**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

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**More Examples Find the derivative: 4. f(x) = x3(x2 + 5)**

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**You Try Find the derivative of each function.**

1. f(x) = x f(x) = x6 – x5 – x4 2. f(x) = x3 + 2x f(x)=(x + 1)(x2 – 2) 3. f(x) = 2x5 – x + 5

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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…

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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

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**Examples Find the antiderivative of each function. 1. f(x) = 3x7**

2. f(x) = 4x2 – 7x + 5 3. f(x) = x(x2 + 2)

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**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)

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Word Problems Page 958, #46

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You Try Page 959, #49

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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.

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Homework Homework 1: Page 958, even Homework 2: page 958, odd, odd

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