AB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 1. Discovering the derivative at x = a: Slopes of secants and tangents to a curve
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1 AB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 1 Discovering the derivative at x = a: Slopes of secants and tangents to a curve 1
2 1. Instantaneous rate of change versus average rate of change Equation of tangent lines. Consider the function y f (x) where f ( x) 3x 2 2 a. Find the average rate of change in f on the interval [-1, 2] b. Find the slope of the tangent line to the graph of f (x) at, i.e. the derivative of f at x = 1. (Precede your answer with Lagrange notation). c. Write an equation of the tangent line to the graph of f (x) at. 2
3 2. Instantaneous rate of change versus average rate of change Let s( t) 5t. Find the average rate of change from t 0 hrs to t 4 hrs. Find the instantaneous rate of change at t 3hrs. 3. Instantaneous rate of change versus average rate of change Let s ( t) t 1. Find the average rate of change from t 0 hrs to t 8hrs. Find the instantaneous rate of change at t 3hrs. 3
4 4. Estimating the derivative with average on a small neighborhood The coordinates f of a moving body for various values of t are given. t (sec) f (ft) Assuming a smooth curve represents the motion of the body, estimate the velocity at t 1. 0 and t 2.5 4
5 5. The derivative at a breaking point x = a. 5 2x; Let f ( x) Find f 1, if it exists. 2 x 1; 5
6 Details and Summary What is the derivative of f at x = a? DEF: The derivative of the function f at the point provided it exists. x a is f f ( a h) f ( a) a) lim h, h ( 0 DEF (Alt): The derivative of the function f at the point provided it exists. x a is f ( a) lim xa f ( x) f ( a), x a Is the function differentiable at x = a? f ( a h) f ( a) DEF: A function f is differentiable at x = a if lim h 0 exists. h If the limit does not exists then we say that the function is not differentiable at x = a. f ( x) f ( a) DEF (alt): A function f is differentiable at x = a if lim xa exists. x a If the limit does not exists then we say that the function is not differentiable at x = a. Theorem: 6
7 AB.Q103.LESSON 1 HW: 1. If f ( 2) 3 and f (2) 5, find an equation of (a) the tangent line, and (b) the normal line to the graph of y f (x) at the point where x Consider the function y f (x) where f ( x) x 2 4x a. Find the average rate of change in f on the interval [-2, 4] b. Find the slope of the tangent line to the graph of f (x) at 1 c. Write an equation of the tangent line to the graph of f (x) at. x, i.e. the derivative of f at x = 1. x; x 0 3. Consider the function y f (x) where f ( x) 2 x x; x 0 a. Find the average rate of change in f on the interval [-2, 4] b. Find the slope of the tangent line to the graph of f (x) at x 0, i.e. the derivative of f at x = 0. c. Write an equation of the tangent line to the graph of f (x) at x Consider the function g( x). x a. Find g (2) using the standard definition of the derivative at x = a. b. Find g (2) using the alternate definition of the derivative at x = a. 5. Use the table below to estimate a) f (1.57) and b) f (3) t f(t)
8 LESSON 2 f ( a h) The Derivative of a Function f at x = a is f ( a) = lim h 0 h f ( a), provided it exists. Right Hand Derivative of f at x = a is f f ( a h) h ( a) lim h0 f ( a), provided it exists. Left Hand Derivative of f at x = a is f f ( a h) h ( a) lim h0 f ( a), provided it exists. f ( a h) f ( a) f ( a) lim or h0 exists when f ( a) f ( a) h (which is important for determining differentiability at the break in a piecewise function) What does it mean that a function fails to be differentiable at x = a? 8
9 EX1. Prove that g is or is not differentiable at x = 0. x 3; x 0 g ( x) 3 2x; x 0 9
10 EX2: Prove that f is or is not differentiable at x = 1. x 1; f ( x) 2 x 3x; 10
11 EX3: Prove that b is or is not differentiable at x = 1. x; b ( x) 2 x; 11
12 3 EX4: Prove that q is or is not differentiable at x = 0. q( x) x 12
13 A function fails the be differentiable at x = a if the following exist at x = a: a discontinuity, corner, cusp, or a vertical line tangent. Corner: Cusp: Vertical Line Tangent: 13
14 LESSON 2 HW: 2 x ; x 0 1. Prove that f ( x) is or is not differentiable at x 0. x; x 0 Classify f (x) as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at x 0. 2; 2. Prove that f ( x) is or is not differentiable at. 2x; Classify f (x) as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at. x; 3. Prove that f ( x) is or is not differentiable at. 2x 1; Classify f (x) as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at 1 x. 3 x ; 4. Prove that f ( x) is or is not differentiable at. 3x; Classify f (x) as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at. x; 5. Prove that f ( x) is or is not differentiable at. 1 x; Classify f (x) as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at 1 x. 6. Prove that f ( x) x x 2 2 is or is not differentiable at x 0. Classify f (x) as either smooth a corner, a cusp, a vertical line tangent, or discontinuous at 0 x. 7. Prove that f ( x) 3 x is or is not differentiable at x 0. Classify f (x) as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at 0 x. 8. Find the unique values of a and b that will make g both continuous and differentiable. 3 x; g ( x) 2 ax bx; 14
15 LESSON 3 Definition: The derivative of the function f with respect to the variable x is the function f ( x h) f ( x) f ( x) whose value at x is f ( x) lim h 0, provided it exists. h 1. Consider the function f ( x) 3x 2 2. A. Find and describe the meaning of f ( x). B. Write an equation of the tangent line to the graph of f(x) at x 1. C. Write an equation of the normal to the graph of f(x) at x 1. D. Find the points on the graph of f where the slope of the tangent line is parallel to y 4x 5. 15
16 2. Graph f ( x) from f (x) The graph of y g(x) shown here is made of line segments joined end to end. Graph the function s derivative. 3. Graph f (x) from f ( x) Sketch a possible graph of a continuous function f that has domain [-3, 3], where f ( 1) 2 and the equation of y f ( x) is shown below. f ( 1; x 1 x) 0; 1 x 2 3; x 2 16
17 LESSON 3 HW: 1. The graph of the function y f (x) shown here is made of line segments joined end to end. Graph y f ( x) and state its domain. 2. Sketch the graph of a continuous function with domain [-2,2], f ( 0) 1, and 1; x 1 f ( x). 2; x 1 3. TEXTBOOK: Section 3.1 #9, 11, 12, 18 17
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