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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.R.5a
Capitolo 4, Problema 4.R.5a

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.

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Identify the critical points of the function f by finding where the derivative f' is equal to zero or undefined. These are the x-values where the graph of f' crosses the x-axis or has discontinuities.
Determine the intervals where the function f is increasing by analyzing the sign of f'. If f' is positive on an interval, then f is increasing on that interval.
Determine the intervals where the function f is decreasing by analyzing the sign of f'. If f' is negative on an interval, then f is decreasing on that interval.
Use the second derivative f" to confirm the nature of the critical points. If f" is positive at a critical point, it indicates a local minimum, and if f" is negative, it indicates a local maximum.
Summarize the findings by listing the critical points and specifying the intervals of increase and decrease for the function f, based on the analysis of f' and f".

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they indicate where the function's slope changes. To find critical points, one typically sets the first derivative, f', equal to zero and solves for the variable.
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Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval where its derivative, f', is positive, and decreasing where f' is negative. By analyzing the sign of the derivative around the critical points, one can determine the intervals of increase and decrease. This information is crucial for understanding the overall behavior of the function.
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Determining Where a Function is Increasing & Decreasing

Second Derivative Test

The second derivative test involves using the second derivative, f'', to determine the concavity of the function at critical points. If f'' is positive at a critical point, the function is concave up, indicating a local minimum; if f'' is negative, the function is concave down, indicating a local maximum. This test helps confirm the nature of critical points found using the first derivative.
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The Second Derivative Test: Finding Local Extrema
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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.

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Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>

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{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

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Rectangles beneath a line


a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

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24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.


ƒ(x) = 4cos (π (x-1)) on [0, 2]

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Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

g(x) = x⁴ - 50x² on [-1, 5]

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