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Limits Section 15-1. WHAT YOU WILL LEARN: 1.How to find the derivatives and antiderivatives of polynomial functions.

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Presentation on theme: "Limits Section 15-1. WHAT YOU WILL LEARN: 1.How to find the derivatives and antiderivatives of polynomial functions."— Presentation transcript:

1 Limits Section 15-1

2 WHAT YOU WILL LEARN: 1.How to find the derivatives and antiderivatives of polynomial functions.

3 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.

4 Remember Our Old Friend Slope? (x+h, f(x+h)) (x, f(x)) Slope of this line Or: h What will happen as “h” gets closer to zero?

5 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.

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

7 Example Find an expression for the slope of the tangent line to the graph of y = x 2 – 4x + 2 at any point. In other words, find. Step 1: find f(x+h): Step 2: find: Step 3: find:

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

9 You Try Find an expression for the slope of the tangent line to the graph of y = 2x 2 – 3x + 4 at any point (find ). Find the slopes of the tangent lines when x = -1 and x = 5.

10 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) = x n, where n is a rational number, then f ’ (x) = nx n-1. Constant Multiple of a Power Rule: If f(x) = cx n, where c is a constant and n is a rational number, then f ’ (x) = cnx n-1. Sum and Difference Rule: If f(x) = g(x) + h(x), then f ’ (x) = g ’ (x) + h ’ (x).

11 Examples Find the derivative of the following: 1. f(x) = x 6 2. f(x) = x 2 – 4x f(x) = 2x 4 – 7x x 2 – 8x – 10

12 More Examples Find the derivative: 4. f(x) = x 3 (x 2 + 5) 5. f(x) = (x 2 + 4) 2

13 You Try Find the derivative of each function. 1. f(x) = x 5 4. f(x) = x 6 – x 5 – x 4 2. f(x) = x 3 + 2x 5. f(x)=(x + 1)(x 2 – 2) 3. f(x) = 2x 5 – x + 5

14 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 x 2 but what about x 2 + 1, x 2 + 2, x …

15 Rules Power Rule: If f(x) = x n, where n is a rational number other than -1, the antiderivative is: Constant Multiple of a Power Rule: If f(x) = kx n, 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

16 Examples Find the antiderivative of each function. 1. f(x) = 3x 7 2. f(x) = 4x 2 – 7x f(x) = x(x 2 + 2)

17 You Try Find the antiderivative of each function. 1. f(x) = 32x 3 2. f(x) = 35x x 2 – 6x f(x) = x 2 (x 2 + x + 3)

18 Word Problems Page 958, #46

19 You Try Page 959, #49

20 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.

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


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