Domanda del libro di testo
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)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 67b
Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line has slope -1/2.
Guida verificata passo dopo passo1
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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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.
Video consigliato:
Percorso guidato
05:44
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.
Video consigliato:
Percorso guidato
05:13Slopes 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.
Video consigliato:
04:50Critical Points
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