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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.10.46b
Chapter 3, Problem 3.10.46b

{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>


b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.

Verified step by step guidance
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First, understand the relationship between the angle of elevation θ and the horizontal distance x from the observer to the point directly below the plane. The angle θ can be expressed using the tangent function: tan(θ) = 400/x, where 400 meters is the height of the plane above the observer.
To find how θ changes with respect to x, differentiate the equation tan(θ) = 400/x with respect to x. This involves implicit differentiation. Recall that the derivative of tan(θ) with respect to θ is sec²(θ) dθ/dx.
Apply implicit differentiation: differentiate both sides of the equation tan(θ) = 400/x with respect to x. The left side becomes sec²(θ) dθ/dx, and the right side becomes -400/x².
Solve for dθ/dx: rearrange the differentiated equation to isolate dθ/dx. This gives dθ/dx = (-400/x²) / sec²(θ). Since sec²(θ) = 1 + tan²(θ), substitute tan(θ) = 400/x into sec²(θ) to express dθ/dx entirely in terms of x.
Graph dθ/dx as a function of x to visualize how the rate of change of the angle θ varies with x. The point at which θ changes most rapidly corresponds to the maximum value of |dθ/dx|. Analyze the graph to determine this point.

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Key Concepts

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

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line from an observer's eye to an object above them and the line of sight to that object. In this context, it helps determine how high the plane appears to the observer as it moves horizontally. Understanding this concept is crucial for analyzing how the angle changes with respect to the plane's position.
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Derivative and Rate of Change

The derivative represents the rate of change of a function with respect to a variable. In this scenario, dθ/dx indicates how the angle of elevation θ changes as the horizontal distance x from the observer to the plane changes. This concept is essential for finding the point where the angle changes most rapidly, which corresponds to the maximum value of the derivative.
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Graphing Functions

Graphing functions involves plotting the relationship between two variables on a coordinate system. For this problem, graphing dθ/dx as a function of x allows us to visualize how the angle of elevation changes with distance. Analyzing the graph helps identify critical points, such as where the angle changes most rapidly, which is key to solving the problem.
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Related Practice
Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

tan xy = x+y; (0,0)

Textbook Question

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.

b. Show that y = B cos t satisfies the equation for any constant B.

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Textbook Question

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

Textbook Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


c. How fast (in fish per year) is the population growing at t=0? At t=5?

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Textbook Question

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 1/√t; a=9, 1/4

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Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→2(x2−3)5−1x−2{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{\left(x^2-3\right)^5-1}{x-2}\)