Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

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1 L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a, b]) is given by Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

2 L15-2 To find the instantaneous rate of change of y with respect to x when x = a: Alternate formula for the instantaneous rate of change of f(x) with respect to x when x = a: lim h 0 f(a + h) f(a) h if the limits exist. = lim b a f(b) f(a) b a We now consider these limits geometrically by looking more closely at slope.

3 Tangent Lines and the Slope of a Curve Consider the graph of a function y = f(x) L15-3 Find the slope of the secant line through P (a, f(a)) and a nearby point Q(a + h, f(a + h)):

4 L15-4 Slope of the Tangent Line Def. The tangent line of the graph of y = f(x) at point P (a, f(a)) is the line through P with slope m = if that limit exists. If not, the graph does not have a tangent line at P. NOTE: We abbreviate to the slope of the curve at (a, f(a)). ex. 1) Find the slope of the tangent line to f(x) = x 2 2x at x = 2.

5 L15-5 2) Find the equation of the tangent line to f(x) = x 2 2x at x = 2. NOTE: ex. Find the slope of the tangent line to f(x) = 3x + 1 at any point (a, f(a)).

6 L15-6 We now give a name to this important limit: Def. Given a function y = f(x). The derivative of f at x = a is defined to be if the limit exists. ex. 1) If f(x) = 1 x + 1, find f ( 3).

7 L15-7 The Derivative as a Function Def. The derivative of function f(x) is defined as f (x) = The derivative is itself a function of x. Its domain: Notations for the Derivative: Given y = f(x) we write The process of finding the derivative is called

8 L15-8 ex. Find f (x) if f(x) = 1 x + 1. Def. A function f is differentiable at x = a if a is in the domain of f and f (a) exists. It is differentiable on an open interval if it is differentiable at each number in the interval. Find each interval for which f(x) = 1 x + 1 is differentiable.

9 ex. Find f (x) if f(x) = 3x 2. L15-9

10 L15-10 Alternate form of the Derivative Function The derivative of f(x) is defined to be NOTE: We have seen two important applications of the derivative: I. For function y = f(x), the tangent line to f(x) at a given x = a is the line through (a, f(a)) whose slope is if it exists. II. The function f (x) represents the instantaneous rate of change of y with respect to x. For economics functions such as cost, revenue and profit, the derivative gives the corresponding marginal cost, revenue or profit. For position function s(t) which gives the displacement of an object from a starting point, the derivative gives instantaneous velocity at a given time.

11 L15-11 ex. Find the equation of the tangent line to f(x) = 1 at x = 3. x + 1 Recall that f (x) = 1 (x + 1) 2.

12 L15-12 Additional Examples ex. Find the x-value of each point at which the tangent line to f(x) = 1 is perpendicular to the x + 1 line y 4x = 6.

13 L15-13 ex. The cost function for a certain product is given by C(x) =.05x x Find and interpret the marginal cost at a production level of 50 items.

14 L15-14 Now you try it! Problems are on pages 14 and Use the graph of f(x) below to estimate, to the nearest multiple of 0.5, the values of f (x) at the points x = 5, 0, Find a limit to answer each question below. 2. Find the slope of the tangent line to f(x) = 4 2 x at any point 3 (x, f(x)) using the limit definition. Does your answer make sense? 3. Find f (x) if f(x) = 10 3x. Then find the slope of the tangent line to f(x) at x = 5 and x = 1. Sketch the graph of f(x) and 3 those tangent lines. Now find f (10/3). What is happening to the slope of the tangent lines to f(x) as x (10/3)? 4. Find the equation of the tangent line to f(x) = x + 1 x 2 at x = Find each point at which the tangent line to f(x) = 2 1 x is perpendicular to the line x + 8y = 4.

15 L The position function of an object (displacement in inches from a starting point) is given by s(t) = t 2 2t 24 where t is measured in seconds. (a) Find the velocity with which the object is traveling at time t = 4. What happens at ten seconds? (b) When does the object first come to a stop? (Hint: what is the velocity at that time?) 7. The demand function for a certain product is given by p(x) = 4 x + 10 and the cost function is C(x) = 2x (a) Find the marginal revenue when x = 12. Interpret your answer. (b) Find the profit function P (x). Use the derivative to find a formula for marginal profit at any production level x. At what production level is profit maximized? Confirm by graphing the profit function.

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