Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question

Chapter 3, Differentiation

In Exercises 51- 54, find the slope of the tangent line to the graph of the logarithmic function at the point (1, 0).

On this educational page, we aim to find the slope of the tangent line to the graph of the logarithmic function at the point (1, 0). The exercises include different variations of logarithmic functions. Below we provide each exercise along with detailed explanations of the associated graphs. ### Exercises **In Exercises 51-54, find the slope of the tangent line to the graph of the logarithmic function at the point (1, 0).** **Exercise 51:** \( y = \ln x^3 \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^3 \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1, 0) is marked on the x-axis. - The x-axis ranges from -2 to 6, and the y-axis ranges from -2 to 4. **Exercise 52:** \( y = \ln x^{3/2} \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^{3/2} \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1, 0) is marked on the x-axis. - The x-axis ranges from -2 to 6, and the y-axis ranges from -2 to 4. **Exercise 53:** \( y = \ln x^2 \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^2 \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1, 0) is marked on the x-axis. - The x-axis ranges from -2 to 6, and the y-axis ranges from -2 to 4. **Exercise 54:** \( y = \ln x^{1/2} \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^{1/2} \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1
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Transcribed Image Text:On this educational page, we aim to find the slope of the tangent line to the graph of the logarithmic function at the point (1, 0). The exercises include different variations of logarithmic functions. Below we provide each exercise along with detailed explanations of the associated graphs. ### Exercises **In Exercises 51-54, find the slope of the tangent line to the graph of the logarithmic function at the point (1, 0).** **Exercise 51:** \( y = \ln x^3 \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^3 \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1, 0) is marked on the x-axis. - The x-axis ranges from -2 to 6, and the y-axis ranges from -2 to 4. **Exercise 52:** \( y = \ln x^{3/2} \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^{3/2} \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1, 0) is marked on the x-axis. - The x-axis ranges from -2 to 6, and the y-axis ranges from -2 to 4. **Exercise 53:** \( y = \ln x^2 \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^2 \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1, 0) is marked on the x-axis. - The x-axis ranges from -2 to 6, and the y-axis ranges from -2 to 4. **Exercise 54:** \( y = \ln x^{1/2} \) - **Graph Explanation:** - The graph displays the function \( y = \ln x^{1/2} \) as a thick black curve. - The tangent line at the point (1, 0) is shown as a thin blue line. - The point (1
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