Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 69
Chapter 3, Problem 69

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x) = 1/x; Q (-2, 4)

Verified step by step guidance
1
Step 1: Identify the point P on the graph of f(x) = \(\frac{1}{x}\) as (a, f(a)) = (a, \(\frac{1}{a}\)).
Step 2: Find the derivative of f(x) = \(\frac{1}{x}\), which is f'(x) = -\(\frac{1}{x^2}\). This represents the slope of the tangent line at any point P.
Step 3: Write the equation of the tangent line at point P using the point-slope form: y - \(\frac{1}{a}\) = -\(\frac{1}{a^2}\)(x - a).
Step 4: Substitute the coordinates of point Q (-2, 4) into the tangent line equation: 4 - \(\frac{1}{a}\) = -\(\frac{1}{a^2}\)(-2 - a).
Step 5: Solve the equation from Step 4 for a to find the x-coordinates of all points P where the tangent line passes through Q.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
9m
Was this helpful?

Key Concepts

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

Tangent Line

A tangent line to a function at a given point is a straight line that touches the graph of the function at that point without crossing it. The slope of the tangent line is determined by the derivative of the function at that point. For the function f(x) = 1/x, the derivative f'(x) = -1/x^2 gives the slope of the tangent line at any point P on the graph.
Recommended video:
05:13
Slopes of Tangent Lines

Finding Points on a Graph

To find points P on the graph of a function where a tangent line passes through a specific point Q, we need to set up an equation that relates the coordinates of P and Q. This involves using the point-slope form of the line equation, which incorporates the slope of the tangent line and the coordinates of point P. Solving this equation will yield the x-coordinates of the points P.
Recommended video:
04:50
Critical Points

Graphing Functions and Tangents

Graphing the function f(x) = 1/x and its tangent lines helps visualize the relationship between the function and the points where the tangent lines intersect the point Q. By plotting the function and the calculated tangent lines, one can verify the accuracy of the solutions found algebraically. This graphical representation aids in understanding the behavior of the function and the tangents at various points.
Recommended video:
5:43
Introduction to Tangent Graph
Related Practice
Textbook Question

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = 3x3 + 5x2 + 6x

Textbook Question

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Textbook Question

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line is horizontal.

1
views
Textbook Question

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Textbook Question

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line has slope -1/2.

1
views
Textbook Question

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)