Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1)\). (a) \(y=x^{-1 / 2}\) (b) \(y=x^{-2}\)

Step-by-Step Solution

Answer

The slope of the tangent line at the point (1,1) is -1/2 for \(y=x^{-1/2}\) and -2 for \(y=x^{-2}\).

Step 1: Differentiate the function for case (a)

Apply the power rule for derivatives, which states if given a function \(y=x^n\), its derivative is \(y'=nx^{n-1}\). For part (a), we are given \(y=x^{-1/2}\). The derivative becomes \(y'=-{1/2}x^{-1/2-1}=-{1/2}x^{-3/2}\).

Step 2: Find the slope at point \((1,1)\) for case (a)

Substitute x by 1 to calculate the slope of the tangent line at the point \((1,1)\) for part (a): \(y'(1)=-{1/2}(1)^{-3/2}=-1/2\)

Step 3: Differentiate the function for case (b)

For part (b), the function is \(y=x^{-2}\), applying the power rule for differentiation results in \(y'=-2x^{-2-1}=-2x^{-3}\).

Step 4: Find the slope at point \((1,1)\) for case (b)

Substitute x by 1 to calculate the slope of the tangent line at the point \((1,1)\) for part (b): \(y'(1)= -2(1)^{-3}=-2\).

Key Concepts

Derivative

Understanding derivatives is fundamental in calculus. At its core, a derivative represents how a function changes as its input changes. In simpler terms, if you have a function of a variable, like time or space, the derivative tells you the rate at which something is happening. For example, if we have a function that represents the distance traveled over time, the derivative of this function will give us the speed or velocity—the rate of change of distance with respect to time. Key points about derivatives:

• Represent the slope or rate of change of a function.
• Calculated through differentiation.
• Help to understand the behavior of functions.

Grasping the derivative concept is crucial as it lays the groundwork for complex calculus ideas. Once comfortable with derivatives, you'll find many calculus topics become more approachable.

Tangent line

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. You can think of it as the best linear approximation of the curve around that point. Tangent lines are incredibly insightful in geometry and calculus, providing information about the direction and curvature of the function at a specific location. Important aspects of tangent lines:

• They represent the instant direction of the curve at a point.
• The slope of a tangent line is equal to the derivative at that point.
• They provide a window into understanding local linearity.

Tangent lines bridge the gap between curves and linear functions, offering a simple yet powerful glance at the curve's behavior in a small vicinity.

Power rule

The power rule is a key technique in differentiation and is especially handy for finding derivatives of polynomial functions. The rule simplifies the differentiation process by providing a straightforward formula: if you have a function of the form \( y = x^n \), the derivative is \( y' = nx^{n-1} \). This elegant rule allows you to quickly find how the function behaves as it "grows" in different regions.Characteristics of the power rule:

• Applicable to functions with integer and fractional powers.
• Makes differentiating polynomial functions fast and easy.
• Lays the foundation for understanding more complex derivative rules.

The power rule is an indispensable tool in your calculus toolkit, offering efficiency and clarity when modeling rates of change of polynomial expressions.

Slope of a function

The slope of a function at a specific point is arguably the most intuitive way to understand derivatives. When you talk about the slope, you're discussing how steep the function is at that particular moment. Mathematically, this slope is identical to the value of the derivative of the function at that particular point. Significant insights about the slope of a function:

• A positive slope indicates the function rises as it moves along.
• A negative slope shows the function is descending.
• If the slope is zero, the function is flat at that point (indicative of a peak, trough, or a flat spot).

In calculus, understanding the slope deepens your insight into how functions behave dynamically, whether they describe physical phenomena, finance equations or growth patterns.