Problem 20 - Calculus an Applied Approach

Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=4-x^{2} ;(2,0) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the function \(f(x) = 4 - x^{2}\) at the point (2,0) is -4.

Step 1: Recall the Limit Definition of the Derivative

The limit definition of the derivative \(f'(a)\) of a function \(f(x)\) at a point \(x = a\) is given by: \[f'(a)= \lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\]

Step 2: Substitute the Function and the Given Point

Plug \(f(x) = 4 - x^2\) and the given point $(2, 0)$ into the limit definition of the derivative: \[f'(2)= \lim_{h \to 0}\frac{(4 - (2+h)^{2}) - (4 - 2^{2})}{h}\]

Step 3: Simplify the Expression

Simplify the expression in the numerator and calculate the limit as \(h\) approaches \(0\): \[f'(2)= \lim_{h \to 0}\frac{4 - (4+4h+h^{2}) - 0}{h} =\lim_{h \to 0}\frac{-4h - h^2}{h}\]. This simplifies further to \[f'(2)= \lim_{h \to 0}(-4 - h)\]

Step 4: Evaluate the Limit

Now evaluate the limit as \(h\) approaches \(0\): \[f'(2) = -4 - 0 = -4\]

Key Concepts

Slope of the Tangent Line

Understanding the slope of the tangent line to a curve at a particular point is crucial in calculus. It signifies the rate at which the function value changes in relation to the change in the independent variable, which in graphical terms, is the steepness of the curve at that point. The tangent line is a straight line that touches the curve at one point without crossing it.

For the function f(x) = 4 - x², to find the slope of the tangent line at the point (2, 0), we use the limit definition of the derivative. This action essentially calculates the slope of the tangent line at a particular point by determining how the function behaves as it gets infinitely close to that point, but doesn't go beyond it, representing a snapshot of the function's instantaneous rate of change at that spot.

Derivative Calculation

The derivative of a function at a certain point quantifies how the function value changes as its input changes. It's calculated using the limit definition of the derivative, which is a foundational concept in calculus.

The step-by-step process demonstrated how we applied the definition to the function f(x) = 4 - x² to find its derivative at the point x = 2. It required substituting x + h for x and allowing h to approach zero, which is the process of taking the limit. This rigorous approach not only gives us the slope of the tangent line but also illustrates the concept of differentiation from first principles.

Evaluating Limits

Evaluating limits is essential in the calculation of the derivative. It is the process of finding the value that a function approaches as the input approaches some value.

In our example, after simplifying the expression, we evaluated the limit as h approached 0. This is a fundamental aspect of calculus that allows us to handle situations where direct substitution is not possible due to the creation of undefined expressions such as division by zero. By evaluating the limit, we concluded that the slope of the tangent line at the point (2, 0) is -4, which we determined without actually dividing by zero.

Calculus

Calculus, the mathematical study of continuous change, is a profound tool that combines the concepts of limits, derivatives, integration, and infinite series. In the context of derivatives and the slope of the tangent line, calculus allows us to pinpoint the exact rate of change at any given point along a function's graph.

This introduction to derivatives via the limit definition provides the basis for understanding more complex applications of calculus, such as optimization problems and differential equations. Calculus is instrumental in fields ranging from physics to economics, enabling us to model and analyze dynamic systems with precision.