Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Concavity

Calculus

5. Graphical Applications of Derivatives

Concavity

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2:34 minutes

Problem 126

Textbook Question

Graph f(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2pi. Comment on the behavior of the graph of f in relation to the signs and values of f".

Verified step by step guidance

Step 1: Begin by finding the first derivative of the function f(x) = x cos(x). Use the product rule, which states that if you have a function h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x). Here, u(x) = x and v(x) = cos(x).

Step 2: Calculate the first derivative f'(x). Using the product rule, differentiate u(x) = x to get u'(x) = 1, and differentiate v(x) = cos(x) to get v'(x) = -sin(x). Therefore, f'(x) = 1 * cos(x) + x * (-sin(x)) = cos(x) - x sin(x).

Step 3: Find the second derivative f''(x) by differentiating f'(x) = cos(x) - x sin(x). Differentiate cos(x) to get -sin(x) and use the product rule on -x sin(x) to get -sin(x) - x cos(x). Thus, f''(x) = -sin(x) - sin(x) - x cos(x) = -2sin(x) - x cos(x).

Step 4: Graph the function f(x) = x cos(x) and its second derivative f''(x) = -2sin(x) - x cos(x) over the interval 0 ≤ x ≤ 2π. Use a graphing tool or software to visualize these functions. Observe the points where the second derivative changes sign, as these indicate potential inflection points where the concavity of f(x) changes.

Step 5: Analyze the behavior of the graph of f(x) in relation to the signs and values of f''(x). When f''(x) > 0, the graph of f(x) is concave up, and when f''(x) < 0, it is concave down. Note how these concavity changes correspond to the shape and turning points of the graph of f(x).

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Key Concepts

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

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane to visualize its behavior over a specified interval. For f(x) = x cos x, this means plotting points for x between 0 and 2π and connecting them smoothly. Understanding the shape and key features of the graph, such as intercepts and turning points, is crucial for analysis.

Second Derivative

The second derivative of a function, denoted as f''(x), provides information about the concavity of the function's graph. If f''(x) > 0, the graph is concave up, indicating a local minimum, while f''(x) < 0 suggests concave down, indicating a local maximum. Analyzing the second derivative helps in understanding the acceleration of the function's rate of change.

Behavior of Functions

The behavior of a function refers to how it changes over its domain, including increasing or decreasing trends, and concavity. By examining the signs and values of f(x) and its derivatives, one can infer critical points, inflection points, and overall trends. This analysis is essential for interpreting the relationship between f(x) and its derivatives, especially in the context of graphing.

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Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

f. On what intervals (approximately) is f concave down?

Textbook Question

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

Textbook Question

114. Parabolas

b. When is the parabola concave up? Concave down?

Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

Textbook Question

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

Textbook Question

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

Textbook Question

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

Textbook Question

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \y'' as positive, negative, or zero.

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