Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3
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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 3.10.46a
{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>
a. What is the rate of change of the angle of elevation dθ/dx when the plane is x=500 m past the observer?
Verified step by step guidance1
First, understand the scenario: The plane is flying horizontally at a constant speed of 70 m/s, 400 meters above the observer. The angle of elevation θ is the angle between the line of sight from the observer to the plane and the horizontal line from the observer.
To find the rate of change of the angle of elevation (dθ/dx), we need to relate θ to the horizontal distance x from the observer. Use the tangent function: tan(θ) = opposite/adjacent = 400/x.
Differentiate both sides of the equation tan(θ) = 400/x with respect to time t. Use implicit differentiation: d(tan(θ))/dt = d(400/x)/dt.
Apply the chain rule: sec²(θ) * (dθ/dt) = -400/x² * (dx/dt). Here, dx/dt is the speed of the plane, which is 70 m/s.
Substitute x = 500 m and dx/dt = 70 m/s into the differentiated equation to solve for dθ/dt. Remember that sec²(θ) = 1/cos²(θ), and you can find cos(θ) using the right triangle relationship: cos(θ) = x/√(x² + 400²).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Related Rates
Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the angle of elevation θ changes as the horizontal distance x from the observer changes. This concept is essential for applying implicit differentiation to relate the rates of change of different variables.
Recommended video:
04:16
Intro To Related Rates
Trigonometric Functions
Trigonometric functions, particularly tangent in this context, relate the angle of elevation to the opposite and adjacent sides of a right triangle. The angle θ can be expressed as θ = arctan(opposite/adjacent), where the opposite side is the height of the plane and the adjacent side is the horizontal distance from the observer. Understanding these relationships is crucial for deriving the necessary equations.
Recommended video:
6:04Introduction to Trigonometric Functions
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the variables are not isolated. In this scenario, we will differentiate the equation relating θ, x, and the height of the plane to find dθ/dx. This method allows us to find the rate of change of the angle of elevation with respect to the horizontal distance without explicitly solving for θ.
Recommended video:
05:14Finding The Implicit Derivative
Related Practice
Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = -3x2 - 5x + 1; P(1,-7)
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Textbook Question
Derivatives using tables Let and . Use the table to compute the following derivatives.
<IMAGE>
a.
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Textbook Question
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
x+y³−y=1; x=1
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Textbook Question
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x⁴-x²y+y⁴=1; (−1, 1)
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Textbook Question
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
(x²+y²)²=25/4 xy²; (1, 2)
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