Explanation Given: The function y = x − 4 x Key concepts used: Critical values of a function can be found by solving the equation f ' = 0 . The sign of f ' is positive when graph is increasing. The sign of f ' is negative when graph is decreasing. The sign of f ' ' is positive when graph is concave up. The sign of f ' ' is negative when graph is concave down. Maximum of a function occurs when derivative changes its sign from positive to negative. Minimum of a function occurs when derivative changes its sign from negative to positive. Inflection point occurs when double derivative changes its sign, provided the original function is defined at that point. Calculation: Let us first find critical values of the given function. For that, first we find the derivative of the function and set it equal to zero: ⇒ y = x − 4 x ⇒ y ' = 1 − 4 2 x ⇒ y ' = 1 − 4 2 x = 0 ⇒ 2 x = 4 ⇒ 2 x = 4 ⇒ x = 2 ⇒ x = 4 ⇒ x = 4 Thus, the critical value of the given function is x = 4 Function is increasing when derivative is positive: ⇒ y ' > 0 ⇒ 1 − 4 2 x > 0 ⇒ x ∈ ( 4 , ∞ ) Function is decreasing when derivative is negative: ⇒ y ' < 0 ⇒ 1 − 4 2 x < 0 ⇒ x ∈ ( − ∞ , 4 ) But domain of the function is [0, ∞ )

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.5, Problem 27E

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Chapter 4.5, Problem 26E

Chapter 4.5, Problem 28E

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For the given function, find (a) the equation of the secant line through the points where x has the given values and (b) the equation of the tangent line when x has the first value. y= f(x) = x²+x; x=-1,x=2 a. Which of the following formulas can be used to find the slope of the secant line? ○ A. 2-(-1) f(2) f(-1) 2+(-1) C. 1(2)+(-1) The equation of the secant line is 1(2)+(-1) О в. 2+(-1) f(2)-(-1) D. 2-(-1)

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Use technology to find f'(4), f'(16), f'(-5) for the given function when the derivative exists. f(x) = -2x² + +10x

Chapter 4 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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To graph: The given function.

Solution Summary: The author explains how critical values of a function can be found by solving the equation f'=0.

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To graph: The given function.

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Chapter 4 Solutions

To graph:

The given function.