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Chapter 1: Problem 17

Find an equation of the tangent line to the graph of \(f(x)=x^{3}\) at (a) (-2,-8) (b) (0,0) (c) (4,64)

Short Answer

Expert verified
Tangent lines are: (a) \(y = 12x + 16\), (b) \(y = 0\), (c) \(y = 48x - 128\).

Step by step solution

01

Find the Derivative of the Function

The first step in finding the equation of the tangent line is to determine the derivative of the function. Here, the function is given as \( f(x) = x^3 \). The derivative of this function, which represents the slope of the tangent, is found using the power rule: \( f'(x) = 3x^2 \).
02

Evaluate the Derivative at the Given Points

For each point, we need to find the slope by evaluating the derivative at the x-value of each point. (a) For \((-2, -8)\), evaluate \(f'(-2) = 3(-2)^2 = 12\).(b) For \((0, 0)\), evaluate \(f'(0) = 3(0)^2 = 0\).(c) For \((4, 64)\), evaluate \(f'(4) = 3(4)^2 = 48\).
03

Use Point-Slope Form to Write the Equation of the Tangent Line

The formula for the equation of a line given a slope \(m\) and a point \((x_1, y_1)\) is the point-slope form: \( y - y_1 = m(x - x_1) \). (a) For \((-2, -8)\), \( m = 12 \): \( y + 8 = 12(x + 2) \) which simplifies to \( y = 12x + 16 \).(b) For \((0, 0)\), \( m = 0 \): \( y - 0 = 0(x - 0) \) which simplifies to \( y = 0 \).(c) For \((4, 64)\), \( m = 48 \): \( y - 64 = 48(x - 4) \) which simplifies to \( y = 48x - 128 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative Calculation
To understand how to find a tangent line to a curve, we first need to calculate the derivative of the function involved. For our problem, the function is given as \( f(x) = x^3 \). The derivative, \( f'(x) \), represents the slope of the curve at any given point. How do we calculate this derivative? By using differentiation rules!
Power Rule
The Power Rule is a fundamental tool for finding derivatives. It states that if you have a function \( f(x) = x^n \), where \( n \) is any constant, then the derivative is \( f'(x) = nx^{n-1} \). It helps simplify complex calculations.
In our example of \( f(x) = x^3 \), we lower the power of \( x \) by one and multiply by the original power: \[ f'(x) = 3x^{3-1} = 3x^2 \]. This derivative \( 3x^2 \) gives us the slope of the tangent line at any point on \( f(x) \).
These rules are not just applicable to this function but a wide range of polynomial expressions.
Point-Slope Form
Once we calculate the slope using the derivative, the next step is to write the equation of the tangent line by using the Point-Slope Form. This form is particularly useful because it allows you to write the equation of a line if you know the slope and any point on the line.
  • The Point-Slope Form formula is: \( y - y_1 = m(x - x_1) \).
  • Here, \( m \) is the slope, and \((x_1, y_1)\) is a point on the line.
For instance, at the point \((-2, -8)\), with a slope \(m = 12\), the equation becomes \( y + 8 = 12(x + 2) \).
Understanding this form is crucial because it connects slope and points directly, simplifying the process of writing linear equations.
Slope of a Tangent Line
The slope of a tangent line to a curve at a particular point indicates how the curve behaves at that point. The derivative function we derived earlier, \( f'(x) = 3x^2 \), gives us the slope of the tangent line for any value of \( x \).
  • The slope tells us whether the function is increasing or decreasing at that point.
  • For example: At \((0, 0)\), the slope is \(0\), indicating a horizontal tangent.
  • At \((-2, -8)\), the slope is \(12\), indicating a steep upward tangent.
  • And at \((4, 64)\), with a slope of \(48\), the tangent is even steeper.
This understanding allows us to predict the direction and steepness of the curve near the points of interest.

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Most popular questions from this chapter

On Earth, an object travels \(4.905 \mathrm{~m}\) after 1 sec of free fall. Thus, by symmetry, an athlete would require 1 sec to jump \(4.905 \mathrm{~m}\) high, and another second to come back down. Is it possible for a person to stay in the air for (have a "hang time" of) 2 sec? Can a person have a hang time of 1.5 sec? 1 sec? What do you think is the longest possible hang time achievable by humans jumping from level ground?

The equation $$ S(r)=\frac{1}{r^{4}} $$ can be used to determine the resistance to blood flow, \(S\). of a blood vessel that has radius \(r\), in millimeters (mm). a) Find the rate of change of resistance with respect to \(r\), the radius of the blood vessel. b) Find the resistance at \(r=1.2 \mathrm{~mm}\). c) Find the rate of change of \(S\) with respect to \(r\) when \(r=0.8 \mathrm{~mm}\)

Let \(f\) and \(g\) be differentiable over an open interval containing \(x=a\). If $$ \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\frac{0}{0} \quad \text { or } \quad \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\frac{\pm \infty}{\pm \infty} $$ and if \(\lim _{x \rightarrow a}\left(\frac{f^{\prime}(x)}{g^{\prime}(x)}\right)\) exists, then $$ \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\lim _{x \rightarrow a}\left(\frac{f^{\prime}(x)}{g^{\prime}(x)}\right) $$ The forms \(0 / 0\) and \(\pm \infty / \pm \infty\) are said to be indeterminate. In such cases, the limit may exist, and l'H么pital's Rule offers a way to find the limit using differentiation. For example, in Example 1 of Section \(1.1,\) we showed that $$ \lim _{x \rightarrow 1}\left(\frac{x^{2}-1}{x-1}\right)=2 $$ Since, for \(x=1,\) we have \(\left(x^{2}-1\right)(x-1)=0 / 0,\) we differentiate the numerator and denominator separately, and reevaluate the limit: $$ \lim _{x \rightarrow 1}\left(\frac{x^{2}-1}{x-1}\right)=\lim _{x \rightarrow 1}\left(\frac{2 x}{1}\right)=2 $$ Use this method to find the following limits. Be sure to check that the initial substitution results in an indeterminate form. $$ \lim _{x \rightarrow 2}\left(\frac{x^{3}+5 x-18}{2 x^{2}-8}\right) $$

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=-x^{3}+1 $$

Find the simplified difference quotient for each function listed. $$ f(x)=x^{4} $$
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