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To Sketch the Graphs of functions using Derivatives. Problem. Consider the function f(x) = 8x - x whose derivative is f(x) = 16x 3x Values of x, f(x) and f^i(x) are given in the following table. 6 9 12 15 f(x) = 8x - x f(x) = 16x 3x

To Sketch the Graphs of functions using Derivatives. Problem. Consider the function f(x) = 8x - x whose derivative is f(x) = 16x 3x Values of x, f(x) and f^i(x) are given in the following table. 6 9 12 15 f(x) = 8x - x f(x) = 16x 3x

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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USING DERIVATIVES IN CURVE SKETCHING To Sketch the Graphs of functions using Derivatives. Problem. Consider the function f(x) = 8x - x whose derivative is f(x) = 16x 3x Values of x, f(x) and f^i(x) are given in the following table. 3 9 12 15 8x - x 3 f(x) f(x) = 16x 3x Solve this problem by using the following example on the other picture
Transcribed Image Text:USING DERIVATIVES IN CURVE SKETCHING To Sketch the Graphs of functions using Derivatives. Problem. Consider the function f(x) = 8x - x whose derivative is f(x) = 16x 3x Values of x, f(x) and f^i(x) are given in the following table. 3 9 12 15 8x - x 3 f(x) f(x) = 16x 3x Solve this problem by using the following example on the other picture
USING DERIVATIVES IN CURVE SKETCHING Objective: To sketch the graphs of functions using derivatives. Problem: Consider the function f(x) = 6x -r, whose derivative is f'(x) = 6-2x. Values of x. f(x), and f'(x) are given in the following table. 4 8 9. 8 T) - 6-2 6 0 - 19 - (x)/ 2. -2 -4. -6 Much can be leamed about the graph of f(x) by plotting each point (x. f(x) from the table and then drawing the tangent to the curve at cach point plotted. The tangent in cach case is the line through (x, f(x)) that has stope f'(x). Figure (a) shows a smooth curve drawn through the points and having the corresponding tungent lines. The smooth curve is the graph of f(x). r(3) = 0 f) <0 In figure (a), notice that as x increases, the curve rises whenever f'(x) > 0, and the curve falls whenever f'(x) < 0. When f'(r) = 0, the curve has a horizontal tangent and f(x) attains its maximum value. The observations that we have so far made for polynomial functions are valid for any function f(x) that has a derivative on an interval 1: 1. Iff(x) > 0 on an interval 1, then the graph of f(x) rises as x increases. 2. If r(2) < 0 on an interval I, then the graph of f(x) rises as x increases. 3. If f'() = 0, then the graph f(x) has a horizontal tangent x = c. The funetion may have a local maximum or minimum value, or neither.
Transcribed Image Text:USING DERIVATIVES IN CURVE SKETCHING Objective: To sketch the graphs of functions using derivatives. Problem: Consider the function f(x) = 6x -r, whose derivative is f'(x) = 6-2x. Values of x. f(x), and f'(x) are given in the following table. 4 8 9. 8 T) - 6-2 6 0 - 19 - (x)/ 2. -2 -4. -6 Much can be leamed about the graph of f(x) by plotting each point (x. f(x) from the table and then drawing the tangent to the curve at cach point plotted. The tangent in cach case is the line through (x, f(x)) that has stope f'(x). Figure (a) shows a smooth curve drawn through the points and having the corresponding tungent lines. The smooth curve is the graph of f(x). r(3) = 0 f) <0 In figure (a), notice that as x increases, the curve rises whenever f'(x) > 0, and the curve falls whenever f'(x) < 0. When f'(r) = 0, the curve has a horizontal tangent and f(x) attains its maximum value. The observations that we have so far made for polynomial functions are valid for any function f(x) that has a derivative on an interval 1: 1. Iff(x) > 0 on an interval 1, then the graph of f(x) rises as x increases. 2. If r(2) < 0 on an interval I, then the graph of f(x) rises as x increases. 3. If f'() = 0, then the graph f(x) has a horizontal tangent x = c. The funetion may have a local maximum or minimum value, or neither.
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