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 7

Chapter 3, Problem 7

An equation of the line tangent to the graph of f at the point (2,7) is y = 4x−1. Find f(2) and f′(2).

Verified step by step guidance

Step 1: Understand that the equation of the tangent line at a point (a, f(a)) on the graph of a function f is given by y = f'(a)(x - a) + f(a).

Step 2: Recognize that the given tangent line equation is y = 4x - 1, which can be compared to the general form y = mx + b, where m is the slope of the tangent line.

Step 3: Identify that the slope of the tangent line, m, is 4. This means that f'(2) = 4, as the slope of the tangent line at x = 2 is the derivative of the function at that point.

Step 4: Use the point (2, 7) given in the problem, which lies on both the function f and the tangent line, to find f(2). Since the point is on the tangent line, substitute x = 2 into the tangent line equation: y = 4(2) - 1.

Step 5: Conclude that f(2) = 7, as the y-coordinate of the point (2, 7) is the value of the function at x = 2.

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Key Concepts

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

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function.

Function Value

The function value f(2) refers to the output of the function f when the input is 2. In the context of the tangent line equation provided, we can confirm that f(2) equals 7, as the point (2,7) lies on the graph of the function.

Derivative

The derivative of a function, denoted as f′(x), represents the slope of the tangent line to the graph of the function at any point x. In this case, since the equation of the tangent line is given as y = 4x - 1, the derivative at x = 2 is simply the slope of this line, which is 4.

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Related Practice

Textbook Question

At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.

a. Find an equation relating A to w.

Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

Textbook Question

The volume V of a sphere of radius r changes over time t.

c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?

Textbook Question

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

f(x) = x(x-1)

Textbook Question

The volume V of a sphere of radius r changes over time t.

b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?

Textbook Question

Use differentiation to verify each equation.

d/dx(x / √1−x²) = 1 / (1−x²)^3/2.