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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 67b
Chapter 3, Problem 67b

Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line has slope -1/2.

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Step 1: Find the derivative of the function f(x) = 4\(\sqrt{x}\) - x. The derivative, f'(x), represents the slope of the tangent line at any point x on the graph of f.
Step 2: Set the derivative equal to the given slope of the tangent line, which is -1/2. This will give you an equation to solve for x: f'(x) = -1/2.
Step 3: Solve the equation from Step 2 for x. This will give you the x-coordinates where the slope of the tangent line is -1/2.
Step 4: Substitute the x-values found in Step 3 back into the original function f(x) to find the corresponding y-coordinates. This will give you the points (x, y) on the graph.
Step 5: Verify the points by checking that the derivative at these x-values indeed equals -1/2, ensuring the solution is correct.

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Key Concepts

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

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is represented as f'(x) and provides the slope of the tangent line to the graph of the function at any given point. To find points where the tangent line has a specific slope, we need to compute the derivative and set it equal to that slope.
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Derivatives

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 is equal to the derivative of the function at that point. In this problem, we are looking for points on the graph of f where the slope of the tangent line equals -1/2.
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Slopes of Tangent Lines

Finding Critical Points

Finding critical points involves determining where the derivative of a function is zero or undefined. These points are essential for analyzing the behavior of the function, including identifying where the slope of the tangent line meets specific criteria, such as -1/2 in this case. Solving the equation derived from the derivative will yield the x-values of interest.
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Related Practice
Textbook Question

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)

Textbook Question

Population growth Consider the following population functions.

e.Use a graphing utility to graph the population and its growth rate.

p(t) = 600 (t²+3/t²+9)

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Textbook Question

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

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

Textbook Question

Let f(x) = 4√x - x.

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

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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

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)=x²+1; Q(3, 6)