Week 5: Tangents and Derivatives

Transcription

1 Week 5: Tangents and Derivatives Goals: Introduce tangent line Discuss derivative as a rate of change. Introduce derivative as the limit of a difference quotient. Study some rules for differentiation Suggested Textbook Readings: Chapter 11: Practice Problems: 11.1: 5, 11, 21, 25, : 3, 7, 9 (Use the limit definition of derivatives)

2 Week 5: Tangents and Derivatives 2 Tangent Lines To prepare for this topic, please read section 11.1 in the textbook. Example 1: The graph of y = x 2 is given below. Draw some secant lines P Q to the curve at P = (1, 1) where points Q are given in the following table. Find the slopes of these secant lines. y Q Slope of P Q 7 ( 1, 1) ( 0.5, 0.25) (0, 0) (0.5, 0.25) (0.9, 0.81) (1.1, 1.21) (1.25, ) (1.5,, 2.25) (2, 4) (2.5, 6.25) x

3 Week 5: Tangents and Derivatives 3 (a) Predict the slope of the tangent line of y = x 2 at (1, 1). (b) Verify your prediction algebraically. (c) Find the equation of the line using the slope you obtained. (d) Draw the graph of the line you obtained in (c). y x

4 Week 5: Tangents and Derivatives 4 The tangent line of the graph of a function y = f(x) at point P is defined to be the limiting position of secant lines through point P and a different point Q on the graph as Q approaches P along the curve from both sides. y Q(z, f(z)) Tangent Line Q P (a, f(a)) at P a z x The slope of the tangent line to the curve of the function y = f(x) at the point P (a, f(a)) is f(z) f(a) m = lim z a z a or f(a + h) f(a) m = lim, if z a = h h 0 h provided that the limit exists. Example 2: Find the slope of the tangent line of the curve y = (2, 2). Find the equation of this tangent. x x 1 at the point

5 Week 5: Tangents and Derivatives 5 Average Rate of Change and Instantaneous Rate of Change To prepare for this topic, please read section 11.3 in the textbook. Example 3: A car is moving on the road. At any time t (in seconds) the distance (in meters) it travels is given by s = f(t) = t 2. (a) What is the average velocity from t = 2 to t = 5 (i.e., 2 t 5). (b) Find the average velocities for the time intervals given in the following table. Time Interval Length of Time Interval Displacement Average Velocity t s = f(t + t) f(t) s t 1 t t t t 2 2 t t t 3 2 t 5 (c) What is the velocity of the car at t = 2 second suggested by the above table? Verify it algebraically.

6 Week 5: Tangents and Derivatives 6 The average rate of change of the function y = f(x) over the interval [a, a + x] is average rate of change = change in y change in x = y x = f(a + x) f(a) x The average rate of change is the slope of the secant line between a and a + x on the graph of f(x). The instantaneous rate of change of the function y = f(x) with respect to x at x = a is the limit of the average rates of change as x approaches a: y instantaneous rate of change = lim x 0 x = lim f(a + x) f(a) x 0 x The instantaneous rate of change at x = a is the slope of the tangent line at x = a on the graph of f(x).

7 Week 5: Tangents and Derivatives 7 Derivatives To prepare for this topic, please read section 11.1 in the textbook. The derivative of a function y = f(x) at x = a is defined by f (a) = lim h 0 f(a + h) f(a) h providing that the limit exists. If f (a) exists, f is said to be differentiable at a. Example 4: If f(x) = x 2, find the derivative of f at x = 1, i.e., f (1). The derivative of the function y = f(x) is the function denoted by f (x) and defined by f (x) = lim h 0 f(x + h) f(x) h providing that the limit exists. A function is said to be differentiable on an interval if it is differentiable at every member in the interval. Other common ways to denote the derivative of a function y = f(x) are dy dx, d f(x), y dx [Textbook, Page 484 (12th), Page 495 (13th)] Example 5: Use the definition to find the derivative of y = mx + b. Geometrically the derivative of f at x = a is the slope of the tangent line of the graph of f at x = a.

8 Week 5: Tangents and Derivatives 8 Example 6: Let f(x) = x. (a) Find f (2), f ( 2). (b) Is the function f differentiable at x = 0? (c) Sketch the graphs of f(x) and f (x).

9 Week 5: Tangents and Derivatives 9 Differentiable Functions The function y = x 2 is differentiable on R. The function y = x is not differentiable at x = 0. All power functions, exponential and logarithmic functions are differentiable on their domains. The function f is not differentiable at x = a if f is not continuous at a, the graph of f has a corner at a, or the function has a vertical tangent at a. Example 7: Sketch graphs of some examples of above cases. If f is differentiable at a, then f is continuous at a. However continuity does not imply differentiability. Read Example 7 and 8 in Section 11.1.

10 Week 5: Tangents and Derivatives 10 The derivative of f(x) at x = a is defined as the slope of the tangent line of the graph of y = f(x) at a. So we can estimate slopes of f(x) at some points and sketch the graph of the derivative y = f (x). Example 8: (Obtaining the graph of f from that of f) The following is a graph of a function f. On the second set of axes, sketch the graph of f. y x y x

11 Week 5: Tangents and Derivatives 11 Increasing and Decreasing functions The function y = f(x) is increasing on an interval I if f(x 1 ) < f(x 2 ) whenever x 1 < x 2 in I. The function y = f(x) is decreasing on an interval I if f(x 1 ) > f(x 2 ) whenever x 1 < x 2 in I. [Textbook, Page 567 (12th), Page 577 (13th)] The derivative and the increasing and decreasing nature of f(x) If f (x) > 0 on an interval I (positive slope), then f(x) is increasing over I. If f (x) < 0 on an interval I (negative slope), then f(x) is decreasing over I. [Textbook, Page 568 (12th), Page 578 (13th)]

12 Week 5: Tangents and Derivatives 12 Rules for Differentiation To prepare for this topic, please read section 11.2 in the textbook. How can we obtain formulae for derivatives of functions? There is only one way: the derivative definition in terms of the difference quotient. f (x) = lim h 0 f(x + h) f(x) h Now we use this definition to prove some rules that give us completely mechanical and efficient procedures for differentiation. Example 9: Let f(x) = C where C is a constant. Find f (x). Example 10: Let f(x) = x n, Find f (x) under the following condition. (1) If n is a positive integer.

13 Week 5: Tangents and Derivatives 13 (2) If n is a negative integer. (3) If n is a fraction. Derivative of a Constant If C is a constant, then d dx = 0 Power Rule If n is any real number, then d dx xn = nx n 1

14 Week 5: Tangents and Derivatives 14 Example 11: Differentiate the following functions (1) f(x) = x 7 (2) f(x) = x 3 x 5 (3) f(x) = 1 x