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Chapter 2: Problem 109

Tangent Line Find the equation(s) of the tangent line(s) to the graph of the curve \(y=x^{3}-9 x\) through the point \((1,-9)\) not on the graph.

Short Answer

Expert verified
The equations of the tangent lines to the curve \(y=x^{3}-9x\) through the point (1,-9) not on the graph are \(y = -6x -3\) and \(y = 18x -27\).

Step by step solution

01

Differentiate the curve function

The derivative of the curve function \(y = x^{3} - 9x\) is obtained using the power rule for differentiation. Therefore, \[y' = 3x^{2} - 9\]
02

Find the slope of tangent line

The equation of a line is given by \(y = mx + c\). The value of 'm' gives us the slope of line, which is the derivative of the function at a given point. The slope of the tangent line can be found by setting \(y' = m\). Hence, \[m = 3x^{2} - 9\]
03

Find the equation of tangent line

Now, we plug the point \((1, -9)\) into the equation of the line. We can solve this equation for \(c\), the y-intercept. Hence, \[-9 = m*1 + c\rightarrow c = -9 - m\]
04

Formulate and solve quadratic equation

Substitute the value of m from step 2 into \(c = -9 - m\) and simplify to obtain a quadratic equation \[c = -9 - (3x^{2} - 9) = -3x^{2}\] This equation will yield possible x coordinates of points on the graph where the tangent passes through the point \((1, -9)\). Solve this equation to find the x coordinates: \[0 = -3x^{2} - 12x + 9\]. This can be simplified to: \[x^{2} + 4x - 3 = 0\]
05

Find x values

Factoring the quadratic equation, \[(x - 1)(x + 3) = 0\], provides the x-coordinate solutions \(x = 1\) and \(x = -3\).
06

Determine y-intercept

Substituting these x values into the slope equation from step 2 gives the slopes of the tangent lines: \(m_{1} = 3(1)^{2} - 9 = -6\) and \(m_{2} = 3(-3)^{2} - 9 = 18\). Now substitute these slope values and the x values into the equation found in step 3 to get our y intercepts. For \(x = 1\), \(c = -9 - m_{1} = -9 - (-6) = -3\). For \(x = -3\), \(c = -9 - m_{2} = -9 - 18 = -27\).
07

Write the equations of the tangent lines

Finally the equations of the tangent lines can be written using the slope-intercept form \(y = mx + c\). The equations are \(y = -6x -3\) and \(y = 18x -27\) respectively

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Key Concepts

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

Derivative of a Function
Understanding the derivative of a function is like getting a snapshot of how a graph is moving at any given point. It's a mathematical way of finding the rate at which one thing changes with respect to another. For example, if you were tracking a car's speed, the derivative would tell you how quickly the speed is changing at any moment鈥攖he acceleration.

In calculus, the derivative of a function at a point essentially gives us the slope of the tangent line at that point. To find this derivative, we use various rules and techniques, among which the power rule is a common and powerful tool. When we took the derivative of the curve function given by y = x^3 - 9x, using the power rule, we calculated the rate at which y changes as x changes. The derivative, represented by y', changes with x, giving us a new function which tells us the slope of the tangent at any point along the original curve.
Slope of a Line
The concept of the slope of a line is key to visualizing and understanding the behavior of linear functions and, in a wider sense, any function's rate of change at a given point. The slope is essentially the measure of the steepness of a line.

Slope-Intercept Form

One of the simplest forms to represent the equation of a line is the slope-intercept form, y = mx + c. Here, m represents the slope, while c is the y-intercept, the point where the line crosses the y-axis. In the context of our problem, the slope of the tangent line gives the rate at which y is changing relative to x at a specific point, and by evaluating the derivative at a specific x value, we were able to determine the slope of the line that just snugs up against the curve at the points of tangency.
Power Rule for Differentiation
The power rule is a quick and useful tool that helps you find the derivative of functions where the variable x is raised to a power. This rule simply says that if you have a function x^n, where n is any real number, the derivative is n*x^(n-1).

To apply the power rule, as we did in the exercise, multiply the exponent by the coefficient of x and then subtract one from the exponent. For example, in the exercise, the function y = x^3 - 9x had terms x^3 and x which transformed into 3x^2 and 9 respectively after applying the power rule. It helped simplify the complex problem of finding the equation of a tangent line into manageable steps, giving us a clear path to our solution.

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Most popular questions from this chapter

Graphical Analysis In Exercises 81-84, use a graphing utility to graph the function and find the x-values at which f is differentiable. $$f(x)=|x-5|$$

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