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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 5d
Chapter 4, Problem 5d

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
Plot a possible graph of f.

Verified step by step guidance
1
Begin by analyzing the graph of ƒ'. The graph of ƒ' provides information about the slope of the original function ƒ. Where ƒ' is positive, ƒ is increasing, and where ƒ' is negative, ƒ is decreasing.
Identify the critical points on the graph of ƒ'. These are the points where ƒ' = 0 or where ƒ' is undefined. At these points, the graph of ƒ may have local maxima, minima, or points of inflection.
Examine the graph of ƒ". The second derivative, ƒ", gives information about the concavity of ƒ. If ƒ" is positive, the graph of ƒ is concave up, and if ƒ" is negative, the graph of ƒ is concave down.
Use the information from ƒ' and ƒ" to sketch the graph of ƒ. Start by plotting the critical points and inflection points, then draw the curve ensuring it increases or decreases according to ƒ' and is concave up or down according to ƒ".
Finally, refine the sketch by ensuring continuity and smoothness, and check that the behavior of the graph of ƒ aligns with the given graphs of ƒ' and ƒ". Adjust the sketch as necessary to accurately reflect the derivative information.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative and Its Graph

The derivative of a function, denoted as f', represents the rate of change of the function at any given point. The graph of f' indicates where the original function f is increasing or decreasing. If f' is positive, f is increasing; if f' is negative, f is decreasing. Understanding the behavior of f' is crucial for sketching the graph of f.
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Graphing The Derivative

Second Derivative and Its Graph

The second derivative, denoted as f'', measures the rate of change of the first derivative. It provides information about the concavity of the function f. If f'' is positive, the graph of f is concave up, indicating a local minimum; if f'' is negative, the graph is concave down, indicating a local maximum. This concept is essential for determining the shape of the graph of f.
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The Second Derivative Test: Finding Local Extrema

Critical Points and Inflection Points

Critical points occur where the first derivative f' is zero or undefined, indicating potential local maxima or minima. Inflection points occur where the second derivative f'' changes sign, indicating a change in concavity. Identifying these points is vital for accurately plotting the graph of f, as they highlight significant features of the function's behavior.
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Critical Points
Related Practice
Textbook Question

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>


f. Sketch one possible graph of f.

Textbook Question

For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.


a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .


b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.



ƒ(x) = x⁵/16 ; [-2, 2] <IMAGE>

Textbook Question

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.


g. Use your work in parts (a) through (f) to sketch a graph of ƒ .

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→2 (x² - 2x / (x² - 6x + 8) 

Textbook Question

Let ƒ(x) = (x - 3) (x + 3)²


g. Use your work in parts (a) through (f) to sketch a graph of ƒ.

Textbook Question

Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpital’s Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.