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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 62b
Chapter 3, Problem 62b

Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = 2x2 / (3x - 1); a = 1

Verified step by step guidance
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Step 1: Identify the function and the point of tangency. The function given is \( y = \frac{2x^2}{3x - 1} \) and the point of tangency is at \( x = 1 \).
Step 2: Find the derivative of the function to determine the slope of the tangent line. Use the quotient rule: if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = 2x^2 \) and \( v = 3x - 1 \).
Step 3: Calculate \( u' \) and \( v' \). For \( u = 2x^2 \), \( u' = 4x \). For \( v = 3x - 1 \), \( v' = 3 \). Substitute these into the quotient rule formula to find \( y' \).
Step 4: Evaluate the derivative at \( x = 1 \) to find the slope of the tangent line. Substitute \( x = 1 \) into the derivative \( y' \) to get the slope \( m \).
Step 5: Use the point-slope form of a line to write the equation of the tangent line. The point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in Step 4, and \( (x_1, y_1) \) is the point \( (1, y(1)) \). Calculate \( y(1) \) using the original function.

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Key Concepts

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

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between variables. In this case, the function y = 2x² / (3x - 1) represents a rational function, which can exhibit various behaviors such as asymptotes and intercepts. Understanding how to graph this function is essential for analyzing its shape and identifying key features.
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Graph of Sine and Cosine Function

Tangent Lines

A tangent line to a curve at a given point represents the instantaneous rate of change of the function at that point. It is defined by the derivative of the function evaluated at that point. For the function y = 2x² / (3x - 1), finding the tangent line at x = 1 requires calculating the derivative and using the point-slope form of a line.
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Slopes of Tangent Lines

Derivatives

The derivative of a function measures how the function's output changes as its input changes, providing a way to determine slopes of tangent lines. For the function y = 2x² / (3x - 1), applying the quotient rule will yield the derivative, which is crucial for finding the slope of the tangent line at the specified point. Understanding derivatives is fundamental in calculus for analyzing function behavior.
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Derivatives
Related Practice
Textbook Question

{Use of Tech} Equations of tangent lines

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = e^x; a = ln 3

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Textbook Question

Calculate the derivative of the following functions.

y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Textbook Question

Find an equation of the line tangent to the given curve at a.

y = 2x2 / (3x - 1); a = 1

Textbook Question

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y=3x−4; P(1, −1)

Textbook Question

Let f(x) = x2 - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 0.

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Textbook Question

Let f(x) = x2 - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 2.

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