Question

Find the equation of the line tangent to the graph of $f$ at $(1,1),$ where $f$ is given by $f(x)=2 x^{3}-2 x^{2}+1$.

Name: Problem 69, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-16T17:37:25-08:00
Duration: 1 min 56 s
Channel: Gregory Higby
Description: Find the equation of the line tangent to the graph of f at (1,1), where f is given by f(x)=2 x^3-2 x^2+1.

   Find the equation of the line tangent to the graph of $f$ at $(1,1),$ where $f$ is given by $f(x)=2 x^{3}-2 x^{2}+1$.

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 69

Instant Answer

Solved by Expert Gregory Higby

04/16/2021

Step 1

The derivative of a function gives us the slope of the tangent line at any point. The derivative of $f(x)$, denoted as $f'(x)$, is found by applying the power rule, which states that the derivative of $x^n$ is $n*x^{n-1}$. So, $f'(x) = 6x^{2}-4x$. Show more…

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Find the equation of the line tangent to the graph of $f$ at $(1,1),$ where $f$ is given by $f(x)=2 x^{3}-2 x^{2}+1$.

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Key Concepts

Point-Slope Form

The point-slope form is a method for writing the equation of a line when given a point on the line and its slope. It is useful in forming the equation of a tangent line by substituting the specific point of tangency and the slope obtained from the derivative.

Tangent Line

A tangent line to a curve at a specific point is a straight line that just 'touches' the curve at that point and has the same instantaneous rate of change as the curve there. It provides a linear approximation of the curve near the point of tangency.

Derivative

The derivative of a function at a given point represents the slope of the tangent line to the function at that point. It is a fundamental concept in calculus used to determine the instantaneous rate of change of the function.

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