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Match the graph of each function in (a)–(d) with the graph of its derivative in I-IV. Give reasons for your choices. (a) (b) (c) yA (d)

Match the graph of each function in (a)–(d) with the graph of its derivative in I-IV. Give reasons for your choices. (a) (b) (c) yA (d)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
### Matching Functions with Their Derivatives **Instructions:** Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Provide reasons for your choices. #### Function Graphs: - **(a)**: A smooth curve with a maximum and minimum, resembling a cubic function. The curve starts to the bottom left, rises to the maximum, decreases to the minimum, and then rises again. - **(b)**: A piecewise linear function shaped like an absolute value function, with a sharp point at the origin. - **(c)**: A concave up parabola opening upwards, resembling a quadratic function. - **(d)**: A concave down parabola opening downwards. #### Derivative Graphs: - **I**: A complex curve with inflection points, starting negative, rising to a point above the x-axis, dipping below, and rising again. - **II**: A smooth, concave down parabola. - **III**: An S-shaped curve, resembles the derivative of a cubic function, with changes in direction. - **IV**: Piecewise constant, resembling horizontal lines with a jump discontinuity at the origin. #### Explanations: - **Graph (a)** likely matches **Derivative III**, which is typical for a cubic function because it has an S-shaped derivative representing changes in concavity and slope. - **Graph (b)** likely matches **Derivative IV**, showing a piecewise constant derivative. This is characteristic of an absolute value function, which has a constant positive or negative slope except at the vertex. - **Graph (c)** likely matches **Derivative I**, where the function steadily increases or decreases and has an inflection point which is mirrored in the change of slope. - **Graph (d)** likely matches **Derivative II**, typical for a downward-opening quadratic function, as its derivative would be a linear function with constant negative slope. These explanations are based on the behavior of the graphs in relation to their derivatives, identifying where the slopes are zero, visualizing the changes in direction, and recognizing points of inflection.
Transcribed Image Text:### Matching Functions with Their Derivatives **Instructions:** Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Provide reasons for your choices. #### Function Graphs: - **(a)**: A smooth curve with a maximum and minimum, resembling a cubic function. The curve starts to the bottom left, rises to the maximum, decreases to the minimum, and then rises again. - **(b)**: A piecewise linear function shaped like an absolute value function, with a sharp point at the origin. - **(c)**: A concave up parabola opening upwards, resembling a quadratic function. - **(d)**: A concave down parabola opening downwards. #### Derivative Graphs: - **I**: A complex curve with inflection points, starting negative, rising to a point above the x-axis, dipping below, and rising again. - **II**: A smooth, concave down parabola. - **III**: An S-shaped curve, resembles the derivative of a cubic function, with changes in direction. - **IV**: Piecewise constant, resembling horizontal lines with a jump discontinuity at the origin. #### Explanations: - **Graph (a)** likely matches **Derivative III**, which is typical for a cubic function because it has an S-shaped derivative representing changes in concavity and slope. - **Graph (b)** likely matches **Derivative IV**, showing a piecewise constant derivative. This is characteristic of an absolute value function, which has a constant positive or negative slope except at the vertex. - **Graph (c)** likely matches **Derivative I**, where the function steadily increases or decreases and has an inflection point which is mirrored in the change of slope. - **Graph (d)** likely matches **Derivative II**, typical for a downward-opening quadratic function, as its derivative would be a linear function with constant negative slope. These explanations are based on the behavior of the graphs in relation to their derivatives, identifying where the slopes are zero, visualizing the changes in direction, and recognizing points of inflection.
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