Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.54b

Chapter 3, Problem 3.6.54b

{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.
b. Graph and interpret the gas mileage m(g)/g.

Verified step by step guidance

First, understand the function m(g) = 50g - 25.8g² + 12.5g³ - 1.6g⁴, which represents the number of miles you can drive with g gallons of gas. This is a polynomial function of degree 4.

To find the gas mileage, which is the miles per gallon, you need to calculate m(g)/g. This involves dividing the function m(g) by g, resulting in a new function: m(g)/g = (50g - 25.8g² + 12.5g³ - 1.6g⁴)/g.

Simplify the expression m(g)/g by dividing each term in the polynomial by g. This gives: m(g)/g = 50 - 25.8g + 12.5g² - 1.6g³.

Graph the simplified function m(g)/g = 50 - 25.8g + 12.5g² - 1.6g³ over the interval 0 ≤ g ≤ 4. This graph will show how the gas mileage changes as the amount of gas in the tank varies.

Interpret the graph: Look for key features such as maximum or minimum points, which indicate the most or least efficient gas mileage. Also, observe the overall trend of the graph to understand how gas mileage is affected by the amount of gas remaining.

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Key Concepts

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

Function Analysis

Understanding the function m(g) is crucial, as it represents the relationship between the gallons of gas remaining and the miles driven. Analyzing this polynomial function involves identifying its behavior, such as its maximum and minimum values, which can be determined through calculus techniques like finding critical points and evaluating the function's limits.

Graphing Rational Functions

To graph the gas mileage m(g)/g, one must understand how to graph rational functions, which are formed by dividing one function by another. This involves determining the domain, identifying asymptotes, and analyzing the behavior of the function as g approaches critical values, such as 0 and 4, to accurately represent the mileage per gallon.

Interpretation of Graphs

Interpreting the graph of m(g)/g requires understanding what the graph represents in the context of fuel economy. This includes analyzing the shape of the graph to determine efficiency at different gas levels, identifying peaks that indicate optimal mileage, and understanding how changes in gas consumption affect overall performance.

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Related Practice

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.

b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

Textbook Question

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>

b. Which runner has the greater average angular velocity?

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Textbook Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counter example.

b. d²/dx² (sin x) = sin x.

Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines

b. Graph the tangent and normal lines on the given graph.

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

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