Textbook Question
Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/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 3.6.12a
Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).
Verified step by step guidance1
Identify the position function s = f(t), which represents the number of ground miles from Seattle at time t hours after take-off.
To find the average velocity over the interval [0, 1.5], use the formula for average velocity: \( \text{Average Velocity} = \frac{f(b) - f(a)}{b - a} \), where a and b are the endpoints of the time interval.
In this problem, a = 0 and b = 1.5, so substitute these values into the formula: \( \text{Average Velocity} = \frac{f(1.5) - f(0)}{1.5 - 0} \).
Determine the values of f(1.5) and f(0) from the position function or the given figure, which represent the distances at t = 1.5 hours and t = 0 hours, respectively.
Substitute the values of f(1.5) and f(0) into the average velocity formula to find the average velocity over the first 1.5 hours of the trip.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Position Function
The position function, denoted as s = f(t), describes the location of an object over time. In this context, it represents the distance of the airliner from Seattle as a function of time since take-off. Understanding this function is crucial for analyzing the motion of the airliner and calculating various rates of change, such as velocity.
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Relations and Functions
Average Velocity
Average velocity is defined as the total displacement divided by the total time taken. For the airliner, it can be calculated by finding the change in position over the specified time interval (0 ≤ t ≤ 1.5 hours). This concept is essential for understanding how fast the plane is moving on average during the first part of its journey.
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Calculus of Motion
The calculus of motion involves using derivatives and integrals to analyze the movement of objects. In this scenario, it helps in determining the average velocity and understanding the relationship between position, velocity, and time. Familiarity with these principles allows for a deeper insight into the dynamics of the airliner's trip.
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Related Practice
Textbook Question
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
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Textbook Question
The line tangent to the curve y=h(x) at x=4 is y = −3x+14. Find an equation of the line tangent to the following curves at x=4.
y = (x²-3x)h(x)
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Textbook Question
The volume V of a sphere of radius r changes over time t.
a. Find an equation relating dV/dt to dr/dt.
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Textbook Question
{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).
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Textbook Question
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)
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