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. $$ f(x)=5-x^{2}, \quad(3,-4) $$

Short answer

-6

Step-by-step solution

Calculating the derivative

Firstly, calculate the derivative of the function \(f(x)\). The derivative of a function gives the slope of the tangent line at any given point. For our function, the derivative \(f'(x)\) would be calculated as: \[f'(x) = -2x\]

Find the slope at the given point

Substitute \(x = 3\) into the calculated derivative. This gives us the slope of the function at the point (3, -4). Thus: \[f'(3) = -2 * 3 = -6\]

Interpretation of the result

The slope of the tangent line to the graph of the function at point (3, -4) is -6

Key concepts

Derivative of a Function

Understanding the derivative of a function is crucial when we want to analyze how a function changes at a certain point. Derivatives can be thought of as the instantaneous rate of change or the slope of a curve at a particular point.

For a function like \( f(x) \), the derivative, typically denoted as \( f'(x) \), provides a formula by which we can calculate this rate. It tells us how quickly the function's output is changing for a small change in the input variable. For example, if we're looking at distance over time, the derivative gives the velocity, the rate of distance change per unit of time.

When we find the derivative of a function, we are doing what is called differentiation. This is a core operation in calculus. Differentiation can involve applying rules such as the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function.

Tangent Line Calculus

In calculus, the concept of a tangent line to a curve at a given point is a linear approximation of the curve near that point. The tangent line touches the curve exactly at one point and its slope equals the slope of the curve at that point.

The importance of the tangent line lies in its slope, which represents how steep or flat the curve is at that exact location. This is precisely what the derivative of a function at a specific value gives us. If you were to zoom in infinitely close on the curve at the point of tangency, the function's curve and the tangent line would become indistinguishable.

For the exercise given, finding the slope of the tangent line at the point (3, -4) on the graph of \( f(x)=5-x^{2} \) involves calculating the derivative and then evaluating it at that specific point.

Calculating Derivatives

Calculating derivatives is a fundamental skill in calculus. It allows us to understand the behavior of functions and their graphs. As shown in our exercise, we applied the differentiation process to the function \( f(x)=5-x^{2} \) using the power rule.

The power rule is one of the most basic rules in differentiation and it states that if we have a function like \( x^n \), where \( n \) is any real number, then its derivative is \( nx^{n-1} \). Thus, applying the power rule to \( -x^{2} \) resulted in the derivative \( f'(x)=-2x \). After finding this formula for the slope, we can find the slope at any point along the curve of the function just by plugging the x-value of the point into the derivative.

In the given problem, once we obtained \( f'(x)=-2x \), we substituted the x-value of the point of interest, which was 3, to find the slope at that point. The straightforward calculation led to a slope of -6, completing our task to find the slope of the tangent line at the point (3, -4).