Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Curve Sketching

Calculus

5. Graphical Applications of Derivatives

Curve Sketching

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2:26 minutes

Problem 3.6.54a

Textbook Question

{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.
a. Graph and interpret the mileage function.

Verified step by step guidance

To graph the mileage function m(g) = 50g - 25.8g^2 + 12.5g^3 - 1.6g^4, first identify the domain of the function, which is 0 ≤ g ≤ 4. This means we will only consider values of g within this interval.

Next, calculate key points of the function within the domain. This includes finding the value of m(g) at the endpoints g = 0 and g = 4, as well as any critical points where the derivative m'(g) = 0, which will help identify local maxima or minima.

To find the critical points, take the derivative of m(g) with respect to g: m'(g) = 50 - 51.6g + 37.5g^2 - 6.4g^3. Set m'(g) = 0 and solve for g to find the critical points within the interval [0, 4].

Evaluate the function m(g) at the critical points and endpoints to determine the behavior of the function. This will help in understanding how the mileage changes as the amount of gas changes.

Plot the calculated points and sketch the graph of m(g) over the interval [0, 4]. Interpret the graph by analyzing how the mileage varies with the amount of gas, noting any points where the mileage is maximized or minimized.

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Key Concepts

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

Function Analysis

Function analysis involves studying the properties and behaviors of mathematical functions. In this context, understanding the mileage function m(g) is crucial, as it describes how mileage varies with the amount of gas g. Analyzing this function includes determining its domain, range, and key features such as intercepts and turning points.

Graphing Techniques

Graphing techniques are essential for visually representing mathematical functions. For the mileage function m(g), creating a graph allows for the interpretation of how mileage changes with varying gas levels. This includes plotting points, identifying the shape of the graph, and recognizing trends such as increases or decreases in mileage.

Critical Points and Interpretation

Critical points are values of g where the function's derivative is zero or undefined, indicating potential maxima, minima, or points of inflection. In the context of the mileage function, finding these points helps in understanding the optimal gas levels for maximum mileage. Interpreting these points provides insights into fuel efficiency and driving strategies.

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Textbook Question

{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.

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