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Chapter 1: Problem 44

The line \(y=a x+b\) is tangent to the graph of \(y=x^{3}\) at the point \(P=(-3,-27) .\) Find \(a\) and \(b.\)

Short Answer

Expert verified
a = 27, b = 54.

Step by step solution

01

Write the equation of the tangent line

Given the tangent line at point P(-3, -27) for the curve y = x^3, we can use the general form of the line: y = ax + b. Let's find a and b.
02

Find the derivative of y = x^3

The derivative of y = x^3 is y' = 3x^2. This gives us the slope of the tangent line at any point x.
03

Calculate the slope at x = -3

Substitute x = -3 into the derivative to find the slope at the specific point: y'(-3) = 3(-3)^2 = 27. So, the slope a is 27.
04

Use point-slope form to find b

With the slope a = 27 and the point P(-3, -27), use the point-slope form y - y1 = a(x - x1) to find b. Substitute into y - (-27) = 27(x - (-3)): y + 27 = 27(x + 3).
05

Solve for the y-intercept b

Convert the equation into slope-intercept form y = 27x + b by expanding and isolating y: y = 27x + 81 - 27, hence b = 54.

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Key Concepts

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

Differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents how a function changes as its input changes. It's a fundamental concept in calculus that helps us determine the slope of a function at any given point. In the exercise, we are given the curve represented by the function \(y = x^3\). To find the slope of the tangent line at any point on this curve, we need to find the derivative of \(y = x^3\). The calculation is straightforward: the derivative of \(x^3\) with respect to \(x\) is \(3x^2\). This derivative tells us the rate of change or the slope of the curve at any given point \(x\).
Slope of Tangent Line
The slope of the tangent line to a curve at a particular point is given by the value of the derivative at that point. In our exercise, we need to find the slope of the tangent line to the curve \(y = x^3\) at the point \((-3, -27)\). Using the derivative we found earlier, \(3x^2\), we substitute \(x = -3\) to get the slope at that point.

So, \[ y'(-3) = 3(-3)^2 = 27 \]

Therefore, the slope of the tangent line at \((-3, -27)\) is 27. This slope, represented by 'a' in the equation of the line \(y = ax + b\), helps us define the behavior of the tangent line at that specific point.
Point-Slope Form
The point-slope form of a line's equation is extremely useful for finding the equation of a tangent line when you know a point on the line and its slope. The point-slope form is given by:

\[ y - y_1 = a(x - x_1) \]

Here, \(a\) is the slope, and \((x_1, y_1)\) is a point on the line. In our problem, we know the slope \(a = 27\) and the point \((-3, -27)\). Substituting these values into the point-slope form, we get:

\[ y - (-27) = 27(x - (-3)) \]

Simplifying, we have:

\[ y + 27 = 27(x + 3) \]

This equation represents the tangent line at the point \((-3, -27)\). The next step is to transform this into the slope-intercept form to find the y-intercept.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is the value of \(y\) when \(x = 0\). To find the y-intercept from the point-slope form of our tangent line equation, we need to simplify it into the slope-intercept form \(y = ax + b\).

Starting from:

\[ y + 27 = 27(x + 3) \]

we distribute and simplify:

\[ y + 27 = 27x + 81 \]

Isolating \(y\), we get:

\[ y = 27x + 81 - 27 \]

So, the equation in slope-intercept form is:

\[ y = 27x + 54 \]

Therefore, the y-intercept \(b\) is 54. This tells us that when \(x = 0\), the value of \(y\) is 54, and it completes our equation of the tangent line: \(y = 27x + 54\).

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Most popular questions from this chapter

Determine which of the following limits exist. Compute the limits that exist. Compute the limits that exist, given that $$\lim _{x \rightarrow 0} f(x)=-\frac{1}{2} \quad\( and \)\quad \lim _{x \rightarrow 0} g(x)=\frac{1}{2}.$$ (a) \(\lim _{x \rightarrow 0}(f(x)+g(x))\) (b) \(\lim _{x \rightarrow 0}(f(x)-2 g(x))\) (c) \(\lim _{x \rightarrow 0} f(x) \cdot g(x)\) (d) \(\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}\)

Determine which of the following limits exist. Compute the limits that exist. Use the limit definition of the derivative to show that if \(f(x)=m x+b,\) then \(f^{\prime}(x)=m.\)

(a) Let \(A(x)\) denote the number (in hundreds) of computers sold when \(x\) thousand dollars is spent on advertising. Represent the following statement by equations involving \(A\) or \(A^{\prime}:\) When \(\$ 8000\) is spent on advertising, the number of computers sold is 1200 and is rising at the rate of 50 computers for each \(\$ 1000\) spent on advertising. (b) Estimate the number of computers that will be sold if \(\$ 9000\) is spent on advertising.

Apply the three-step method to compute the derivative of the given function. $$f(x)=x+3$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=x \sqrt{x}$$
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