Question

Finding Equation(s) of Tangent Line(s) 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

The equations of the tangents are \(y = -x -7\) and \(y = -4x -3\).

Step-by-step solution

Find the derivative of the given function

First, we need to differentiate our function with respect to x: The derivative of \(y=x^{3}-9 x\) is \(y'=3x^{2}-9\). This result gives us the slope of the tangent at any point on the function.

Find the slope of the tangent through the given point

The slope of the line passing through the point \((1,-9)\) and a point on the graph \((x,x^{3}-9x)\) is given by \((x^3 - 9x + 9)/(x-1)\). We can find values of x such that this slope is the same as \(y'\) i.e. solve the equation \((x^3 - 9x + 9)/(x-1) = 3x^{2}-9\).

Solve for x

Solving the equation, we get x = -1, and x = 3 (only real solutions are considered). This gives us the points of tangency as \((-1,-8)\) and \((3,-18)\).

Find the equations of the tangent lines

Finally, we use point-slope form, \(y - y1 = m(x - x1)\), to find the equations of the tangent lines going through the point \((1, -9)\) and points of tangency \((-1,-8)\) and \((3,-18)\). This gives us the equations as \(y = -7 - x\) and \(y = -4x -3\).

Key concepts

Understanding the Derivative

In calculus, the derivative of a function is a measure of how a function's output value changes as its input value changes. When we talk about the derivative, think of it as the slope of the function at any given point.
For the function given in our exercise, which is \( y = x^3 - 9x \), we differentiate it with respect to \( x \) to find the derivative. Differentiation here involves applying basic power rules to get \( y' = 3x^2 - 9 \). This result is crucial because it gives us the formula for the slope of the tangent line to the curve at any point \( x \).
Derivatives are foundational in calculus as they allow us to understand how functions behave, particularly how quickly they increase or decrease over intervals.

Exploring the Tangent Line

A tangent line to a curve at a given point is the straight line that just "touches" the curve at that point. This line has the same slope as the curve at the point of tangency, making it a fantastic tool for understanding the curve's behavior locally.
In our exercise, we're tasked to find the tangent line to the curve \( y = x^3 - 9x \) that passes through a specific external point \( (1, -9) \). The essential step here involves ensuring that the slope of the tangent line matches the derivative at the points of tangency. We derive from the conditions necessary to solve for points where this is true.
Knowing how to work with tangent lines is essential for further complex concepts in calculus, like optimization problems and error approximations.

The Use of Point-Slope Form in Finding Equations

Point-slope form is a straightforward way to express the equation of a line when you know a point on the line and its slope. The general formula is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point and \( m \) is the slope.
In our scenario, once we have identified the points of tangency \((-1, -8)\) and \((3, -18)\) with the respective slopes given by the derivative, we utilize point-slope form to compute the equations of tangent lines passing through these points.
This is extremely helpful in calculus, not just for finding tangent lines, but for linear approximations of curves at small intervals.

Differentiation: The Process and Importance

Differentiation is the process of finding the derivative, which involves mathematical techniques to determine how a function changes. It forms the foundation of calculus, enabling us to solve a wide range of problems.
In differentiation, we commonly use rules like the power rule, product rule, and quotient rule to simplify the process of finding derivatives. For our given function, applying the power rule allows us to quickly find the derivative.
Understanding differentiation is vital as it applies to real-world scenarios, ranging from calculating speed in physics to determining rates of growth in biology or economics.