In Exercises 5-10 , find an equation for the tangent to the curve at the given point. Then sketc

Step 1: First, we need to find the derivative of the given function. The function is $y=1/x^{3}$

In Exercises 5-10 , find an equation for the tangent to the...

Key Concepts

Slope

The slope is a measure of the steepness or the rate at which a line ascends or descends. In the context of calculus, when dealing with curves, the slope at a particular point is given by the derivative of the function at that point. This concept is essential for understanding the instantaneous rate of change and is used to characterize the behavior of the curve.

Graphing Functions and Tangents

Sketching the graph of a function along with its tangent line involves plotting the curve of the function and accurately drawing the line that just touches the curve at the point of interest. This process serves as a useful visual representation of the function's local linear approximation near the point of tangency and reinforces the concept that the derivative provides the slope of the tangent line at that point.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its independent variable. It is a fundamental concept in calculus used to determine the slope of the tangent line at any point on a curve. Calculating the derivative involves applying differentiation rules, which allow us to understand how the function behaves locally around a specific point.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without cutting across it. The slope of the tangent line is equal to the derivative of the function at that point. The equation of the tangent line is typically found by using the point-slope form of a line, incorporating both the coordinates of the point of tangency and the calculated slope.

In Exercises $5-10$ , find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. $$ y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right) $$

Instant Answer

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