Textbook Question
9–61. Evaluate and simplify y'.
y = tan^−1 √t²−1
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.2.45a
Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>
a. At which points is the slope of the curve negative?
Verified step by step guidance1
Step 1: Understand that the slope of a curve at a point is determined by the derivative of the function at that point. A negative slope indicates that the function is decreasing at that point.
Step 2: Identify the points on the graph where the curve is sloping downwards as you move from left to right. These are the points where the slope is negative.
Step 3: Examine the graph visually or use the derivative function if available to determine the slope at each point A, B, C, D, and E.
Step 4: For each point, check if the tangent line to the curve at that point has a negative slope. This can be done by observing if the tangent line is angled downwards from left to right.
Step 5: List the points where the slope is negative based on your observations from the graph or calculations from the derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope of a Curve
The slope of a curve at a given point represents the rate of change of the function at that point. It is determined by the derivative of the function, which indicates how steeply the curve rises or falls. A positive slope means the curve is increasing, while a negative slope indicates it is decreasing.
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Negative Slope
A negative slope occurs when the curve descends as you move from left to right. This means that as the x-values increase, the y-values decrease. Identifying points with a negative slope is crucial for understanding where the function is decreasing, which can be visually assessed on a graph.
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Graphical Analysis
Graphical analysis involves examining the visual representation of a function to determine its characteristics, such as slope. By observing the curve's behavior at various points, one can identify intervals of increase and decrease, as well as specific points where the slope changes from positive to negative or vice versa.
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Related Practice
Textbook Question
Find f′(1) when f(x) = x^(1/x).
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Textbook Question
If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?
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Textbook Question
A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e-t cos t), for t ≥ 0.
Determine her velocity at t = 1 and t = 3.
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Textbook Question
9–61. Evaluate and simplify y'.
y = csc⁵ 3x
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Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = 2/√x; P(4,1)
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