Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 3: Problem 2

Explain why the slope of a secant line can be interpreted as an average rate of change.

Short Answer

Expert verified
Answer: The slope of a secant line can be interpreted as an average rate of change because it represents the change in output (y-values) per unit change in input (x-values) between two distinct points on a curve. This change in output per unit change in input is essentially the average rate of change for the function between the two points.

Step by step solution

01

Define a secant line

A secant line is a straight line that intersects a curve at two distinct points, P and Q. In the context of a function, the two points represent two different inputs (x-values) and their respective outputs (y-values).
02

Calculate the slope of the secant line

To calculate the slope of the secant line, we'll use the formula for the slope of a line, which is given by: slope = (change in y) / (change in x) or m = (y2 - y1) / (x2 - x1) Here, (x1, y1) and (x2, y2) are the coordinates of points P and Q, respectively.
03

Relate the slope to the average rate of change

The slope of the secant line represents the change in y (output) per unit change in x (input) between the two points P and Q. This change in y per unit change in x is essentially the average rate of change for the function between the two points. To see this more clearly, let's denote the function as f(x), the input as x, and the output as y = f(x). Then, the average rate of change between points P and Q would be: average rate of change = (f(x2) - f(x1)) / (x2 - x1) This expression is the same as the formula for the slope of a secant line, which shows that the slope of a secant line can be interpreted as an average rate of change.
04

Provide an example

To illustrate this concept, let's use a simple linear function: f(x) = 2x. We will choose two points on the function's graph: P(1, 2) and Q(3, 6). Now, let's compute the slope of the secant line and the average rate of change between these two points: slope = (y2 - y1) / (x2 - x1) = (6 - 2) / (3 - 1) = 4 / 2 = 2 average rate of change = (f(x2) - f(x1)) / (x2 - x1) = (6 - 2) / (3 - 1) = 4 / 2 = 2 As seen in this example, the slope of the secant line is the same as the average rate of change in the function between points P and Q.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A lighthouse stands 500 m off a straight shore and the focused beam of its light revolves four times each minute. As shown in the figure, \(P\) is the point on shore closest to the lighthouse and \(Q\) is a point on the shore 200 m from \(P\). What is the speed of the beam along the shore when it strikes the point \(Q ?\) Describe how the speed of the beam along the shore varies with the distance between \(P\) and \(Q\). Neglect the height of the lighthouse.

Suppose you forgot the Quotient Rule for calculating \(\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right) .\) Use the Chain Rule and Product Rule with the identity \(\frac{f(x)}{g(x)}=f(x)(g(x))^{-1}\) to derive the Quotient Rule.

Two boats leave a port at the same time, one traveling west at \(20 \mathrm{mi} / \mathrm{hr}\) and the other traveling southwest at \(15 \mathrm{mi} / \mathrm{hr} .\) At what rate is the distance between them changing 30 min after they leave the port?

Let \(f(x)=\cos ^{2} x+\sin ^{2} x\). a. Use the Chain Rule to show that \(f^{\prime}(x)=0\). b. Assume that if \(f^{\prime}=0,\) then \(f\) is a constant function. Calculate \(f(0)\) and use it with part (a) to explain why \(\cos ^{2} x+\sin ^{2} x=1\).

Proof by induction: derivative of \(e^{k x}\) for positive integers \(k\) Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually \(k=1\) ). In the second step, the statement is assumed to be true for \(k=n\), and the statement is proved for \(k=n+1,\) which concludes the proof. a. Show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x},\) for \(k=1\) b. Assume the rule is true for \(k=n\) (that is, assume \(\left.\frac{d}{d x}\left(e^{n x}\right)=n e^{n x}\right),\) and show this assumption implies that the rule is true for \(k=n+1\). (Hint: Write \(e^{(n+1) x}\) as the product of two functions and use the Product Rule.)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free