2 What You Will Learn: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? Slope of this lineOr:(x+h, f(x+h))What will happen as “h” gets closer to zero?(x, f(x))h
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 DifferentiationThe process of finding the derivative is called differentiation.Notation for f’(x) looks like: and is read “dy, dx”
7 ExampleFind an expression for the slope of the tangent line to the graph of y = x2 – 4x + 2 at any point. In other words, findStep 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 TryFind 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.
10 Rules for Finding Derivatives of Polynomials Constant Rule: The derivative of a constant function is 0. If f(x) = c then f’(x) = 0Power 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).
12 More Examples Find the derivative: 4. f(x) = x3(x2 + 5)
13 You Try Find the derivative of each function. 1. f(x) = x f(x) = x6 – x5 – x42. f(x) = x3 + 2x f(x)=(x + 1)(x2 – 2)3. f(x) = 2x5 – x + 5
14 AntiderivativesWe 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…
15 RulesPower 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 off(x) + or – g(x) is
16 Examples Find the antiderivative of each function. 1. f(x) = 3x7 2. f(x) = 4x2 – 7x + 53. f(x) = x(x2 + 2)
17 You Try Find the antiderivative of each function. 1. f(x) = 32x3 2. f(x) = 35x6 + 12x2 – 6x + 123. f(x) = x2(x2 + x + 3)
20 SummaryDerivative 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.