Problem 9 - Calculus of a Single Variable

Question

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ f(t)=3 t-t^{2}, \quad(0,0) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the function \(f(t)=3 t-t^{2}\) at the point (0,0) is 3.

Step 1: Take Derivative

First, the derivative of the function \(f(t)\) needs to be taken. The derivative, \(f'(t)\), of a function is the function that gives the slope of the original function at any point. The derivative of \(f(t)=3 t-t^{2}\) is \(f'(t) = 3 - 2t\).

Step 2: Substitute the x-value of the point into the derivative

Next, the x-value (t-value here) of the given point should be substituted into the derivative. The given point is at \(t=0\). Substituting this into the derivative: \(f'(0) = 3 - 2(0)\).

Step 3: Simplify

Finally, simplify \(f'(0) = 3 - 2(0)\) to find the slope of the tangent line. The resulting slope is 3.

Key Concepts

Derivative Calculation

Calculating the derivative is a fundamental part of finding the slope of a tangent line. The derivative of a function, denoted as \(f'(t)\), provides the slope of the function at any given point. In the provided exercise, the function \(f(t) = 3t - t^2\) needs to be differentiated to find its derivative. This involves applying basic rules of differentiation, such as the power rule.

When using the power rule, each term of the function is differentiated separately. For \(3t\), the derivative is 3, since the power of \(t\) is 1. For \(-t^2\), by applying the power rule, the derivative becomes \(-2t\).

Thus, by combining these results, the derivative of the entire function is \(f'(t) = 3 - 2t\). This expression allows you to calculate the slope of the tangent line at any point \(t\).

Tangent Line

The idea of a tangent line is essential in calculus and describes a line that touches a curve at exactly one point without crossing it. This point is called the point of tangency.

At this specific point, the tangent line has the same slope as the curve itself. Thus, finding the tangent line involves determining the slope of the curve at that particular point.

In the given exercise, the tangent line to the function \(f(t) = 3t - t^2\) at the point \((0,0)\) is of interest. By differentiating the function, we can obtain the expression for its slope and subsequently use it to describe the tangent line at this specific point.

Function Analysis

Function analysis involves understanding a function's behavior by studying its derivative. The derivative provides crucial insights into the function's increasing or decreasing nature over its domain.

By examining the derivative \(f'(t) = 3 - 2t\) of the given function, it is easy to see how the function changes over time. When the derivative is positive, the function is increasing, whereas when it is negative, the function is decreasing.

In the exercise, the derivative at \(t=0\) was found to be 3, indicating that at this point, the function is increasing. This understanding of the function's behavior helps in sketching its graph and analyzing its real-world implications.

Differentiation

Differentiation is the mathematical process used to find the derivative of a function. It involves applying known rules such as the power rule, constant rule, and sum rule.

In the given exercise, the differentiation of \(f(t) = 3t - t^2\) utilizes these rules to generate \(f'(t) = 3 - 2t\). The power rule is applied, where the exponent of \(t\) is decreased by one, and the original power is multiplied by the coefficient.

Differentiation allows us to understand not just slopes, but also rates of change, inflection points, and concavity, making it an invaluable tool in calculus.

Mathematical Slope

In calculus, the concept of slope refers to the measure of the steepness or incline of a line. For a line, slope is constant, but for a curve, the slope can change from point to point.

When dealing with functions, the slope at any point is given by the derivative, providing a moment-by-moment account of how steep the curve is.

In the exercise, the slope of the function \(f(t) = 3t - t^2\) at \(t=0\) is determined using its derivative, \(f'(t) = 3 - 2t\). By substituting the \(t\)-value into this derivative, we find the slope at that particular point, which is 3. This tells us that at \((0,0)\), the tangent line is quite steep, indicating a fairly rapid change at that specific point.