Question

(a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ y=x^{4}-x \quad(-1,2) $$

Short answer

The equation of the tangent line to the function at the point (-1,2) is \(y=-5x-3\).

Step-by-step solution

Find the derivative

Derivative of the function \(y=x^{4}-x\) can be calculated using the power rule \((x^n)' = n \cdot x^{n-1}\) for differentiation. Thus, \(y'=4x^{3}-1\).

Calculate the slope of the tangent line

Next, substitute the x-coordinate of the given point into the derivative to find the slope of the tangent line. For point \((-1,2)\), the slope \(m=y'(-1)=4(-1)^{3}-1=-5\).

Use the point slope form to find the equation of the line

The equation of a line in point-slope form is \(y-y_{1}=m(x-x_{1})\). Here, \(x_{1}=-1\), \(y_{1}=2\), and \(m=-5\). Substituting these values gives the equation \(y-2=-5(x+1)\). Simplifying gives the tangent line equation as \(y=-5x-3\).

Graph the function and its tangent line

Next, use a graphing tool to graph the function \(y=x^{4}-x\) and its tangent line \(y=-5x-3\). It can be seen that the tangent line touches the function graph at the given point \((-1,2)\).

Validate the graph

Use the derivative feature of your graphing tool. The derivative of function should match the slope of the tangent line, validating the steps above.

Key concepts

Derivative of a Function

The derivative of a function is a fundamental concept in calculus that represents the rate at which a function's output value changes with respect to changes in its input value. More specifically, it provides us with the slope of the tangent line at any point along the curve of the function. For example, let's consider the function
\( y = x^4 - x \).
Using the power rule of differentiation, which states that \( (x^n)' = n \times x^{n-1} \), we can calculate its derivative to be
\( y' = 4x^3 - 1 \).
This derivative function now tells us the slope of the tangent line at any point \( x \) on the graph of the function \( y \).

Slope of a Tangent Line

The slope of a tangent line to a curve at a particular point is the rate at which the function is increasing or decreasing at that point. It is the same as the value of the derivative of the function at that point. To find the slope of the tangent line to the function
\( y = x^4 - x \)
at point (-1,2), we substitute \( x = -1 \) into the derivative we found earlier:
\( m = y'(-1) = 4(-1)^3 - 1 = -5 \).
Hence, the slope of the tangent line at this point is -5. A negative slope indicates that the function is decreasing at this point, and a slope of -5 tells us it is decreasing quite rapidly.

Point-Slope Form

When writing the equation of a line, the point-slope form can be extremely helpful, especially when we know a particular point on the line and the slope. The point-slope form is written as
\( y - y_1 = m(x - x_1) \),
where \( (x_1, y_1) \) is the point on the line and \( m \) is the slope of the line. For the tangent line at point (-1,2) with a slope of -5, the equation becomes
\( y - 2 = -5(x + 1) \),
which simplifies to \( y = -5x - 3 \). This is the equation of our tangent line, and it explicitly shows that the line crosses the \( y \) axis at -3 and decreases by 5 units on the \( y \) axis for each unit increase in \( x \) axis.

Graphing Functions

Graphing functions is an essential skill for visualizing relationships between variables. Using a graphing utility, we can easily see the behavior of functions and their tangent lines. When graphing
\( y = x^4 - x \)
and its tangent line
\( y = -5x - 3 \),
we expect the tangent line to touch the graph of the function precisely at the point (-1,2). This visual confirmation can help solidify our understanding of derivatives and slopes. Moreover, graphing tools often have features to calculate derivatives at given points, allowing us to quickly check our work. By confirming the slope found algebraically is the same as the slope from the graphing tool, we validate our understanding of these crucial concepts.