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

4. Applications of Derivatives

Linearization

Calculus

4. Applications of Derivatives

Linearization

Previous problemNext problem

2:21 minutes

Problem 4.6.13

Textbook Question

Estimating speed Use the linear approximation given in Example 1 to answer the following questions.

If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed?

Verified step by step guidance

To estimate the speed, we can use the concept of linear approximation. Linear approximation is a method of estimating the value of a function near a given point using the tangent line at that point.

First, identify the function that relates distance and time. The speed is the derivative of the distance with respect to time. If we denote distance as \( s(t) \) and time as \( t \), then speed \( v(t) \) is given by \( v(t) = \frac{ds}{dt} \).

Given that you travel one mile in 59 seconds, we can set up the function \( s(t) = t \) where \( s(t) \) is in miles and \( t \) is in seconds. The derivative \( \frac{ds}{dt} = 1 \) mile per second.

To find the approximate average speed, convert the time from seconds to hours since speed is typically measured in miles per hour. There are 3600 seconds in an hour, so the time in hours is \( \frac{59}{3600} \) hours.

Finally, calculate the approximate average speed using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). Substitute the values: \( \text{Speed} = \frac{1 \text{ mile}}{\frac{59}{3600} \text{ hours}} \). This will give you the approximate average speed in miles per hour. For the exact speed, use the same formula without approximation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It relies on the idea that for small changes in the input, the function can be approximated by a linear function. This is particularly useful in calculus for estimating values without needing to compute them exactly.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. It provides a measure of how fast an object is moving over a specific interval. In this context, if you travel one mile in 59 seconds, the average speed can be calculated by converting the distance and time into consistent units, typically miles per hour.

Exact Speed

Exact speed refers to the precise measurement of speed at a specific moment or over a specific distance and time. In contrast to average speed, which considers total distance and time, exact speed can be determined using calculus concepts such as derivatives, which provide instantaneous rates of change. This allows for a more accurate representation of speed at any given point.

Watch next

Master Linearization with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

e⁰·⁰⁶

Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

1/³√510

Textbook Question

45–46. Linear approximation

a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?

ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Textbook Question

Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).

Textbook Question

Find the linearizations of

a. tan x at x = -π/4

Graph the curves and linearizations together.

Textbook Question

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

Textbook Question

____

Find the linearization of ƒ(x) = 2/ (1 - x) + √1 + x - 3.1 at x = 0.

Textbook Question

Finding Linearizations

In Exercises 1–5, find the linearization L(x) of f(x) at x = a.