Assigment Wk 5

Calculus: Learning Worksheets Chapter 4 Name ________________________________ Date ______________ Class ____________ Section 4-1 First Derivatives and Graphs Goal: To use the first derivative to analyze graphs Theorem 1: Increasing and Decreasing Functions For the interval (a, b), if f ‘( x)  0, then f ( x) is increasing and the graph of f rises. If f ‘( x)  0, then f ( x) is decreasing and the graph of f falls. Definition: Critical Values The values of x in the domain of f where f ‘( x)  0 or where f ‘( x) does not exist are called the critical values of f. Procedure: First Derivative Test for Local Extrema Let c be a critical value of f. Construct a sign chart for f ‘( x) close to and on either side of c. Then f (c) will be a) A local minimum if f ‘( x) changes from negative to positive. b) A local maximum if f ‘( x) changes from positive to negative. c) Neither a local minimum nor a local maximum if f ‘( x) does not change signs. 1. Given the function f ( x) = 3 x3 − 81x + 13 find a) f ‘( x) b) The critical values of the function c) The partition numbers for the first derivative 125 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 2. Given the function f ( x) = Chapter 4 21 find x+3 a) f ‘( x) b) The critical values of the function c) The partition numbers for the first derivative 126 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 In Problems 3–5, for the functions given, find: a) The intervals on which the function is increasing b) The intervals on which the function is decreasing c) The local extrema 3. f ( x) = 5 x 2 + 20 x − 12 127 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 4. Chapter 4 f ( x) = 3 x3 − 81x + 13 128 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 5. Chapter 4 f ( x) = 3 x 4 + 4 x3 − 8 129 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 In Problems 6–7, for the functions given, find a) The intervals on which the function is increasing b) The intervals on which the function is decreasing c) Sketch the graph of the function 6. f ( x)  2 x3  3 x 2  12 x 130 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 7. Chapter 4 f ( x)  x3  12 x  4 131 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 132 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 Name ________________________________ Date ______________ Class ____________ Section 4-2 Second Derivatives and Graphs Goal: To use the second derivative to analyze graphs Notation: Second Derivative For y  f ( x), the second derivative of f, provided that it exists, is f ”( x)  Definition: d f ‘( x). dx Concavity The graph of a function f is concave upward on the interval (a, b) if f ‘( x) is increasing on (a, b) and is concave downward on the interval (a, b) if f ‘( x) is decreasing on (a, b). Summary: Concavity For the interval (a, b), if f ”( x)  0, then f ‘( x) is increasing and the graph of f is concave upward. If f ”( x)  0, then f ‘( x) is decreasing and the graph of f is concave downward. Theorem: Inflection Point If y  f ( x) is continuous on (a, b) and has an inflection point at x  c, then either f ”(c)  0 or f ”(c) does not exist. Procedure: Graphing Strategy (first version) Step 1. Analyze f ( x). Find the domain and the intercepts Step 2. Analyze f ‘( x). Find the partition numbers for, and the critical values of, f ‘( x), and determine local extrema. Step 3. Analyze f ”( x). Find the partition numbers for f ”( x), and determine concavity. Step 4. Sketch the graph of the function f. 133 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 1. Given the function f ( x) = 7 x 4 − 11x 2 − 15 x + 3, find f ”( x). 18 2. Given the function f ( x) = 4 x5 + 12 x 5 − 19, find f ”( x). 3. Given the function f ( x) = 2 x3 − 18 x 2 + 7 x − 11, find the x and y coordinates of all inflection points. 134 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 In Problems 4–6, for the functions given, find a) The intervals on which the function is concave upward b) The intervals on which the function is concave downward c) The inflection points 4. f ( x) = 2 x 4 − 48 x 2 135 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 5. Chapter 4 f ( x) = 3 x3 − 11x 2 + 4 x − 10 136 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 6. Chapter 4 f ( x) = 16e x − 2e 2 x 137 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 In Problems 7–9, apply the 4 step graphing procedure and sketch the graph of the following functions. 7. f ( x)  ( x  2)( x 2  x  4) 138 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 8. Chapter 4 f ( x)  5 x 4  10 x3 139 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 9. Chapter 4 f ( x)  8e x  e 2 x 140 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 10. A company estimates that it will sell N ( x) units of a product after spending x thousand dollars on advertising, as given by N ( x)  0.25 x 4  23 x3  360 x 2  45, 000 25  x  54. When is the rate of change of sales increasing and when is it decreasing? What is the point of diminishing returns and the maximum rate of change of sales? 141 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 142 Copyright © 2019 Pearson Education, Inc.

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