Problem 39 - Calculus an Applied Approach

Question

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{2} x^{2} ;(2,2) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the graph of the function \(f(x) = \frac{1}{2}x^2\) at the point (2,2) is \(y = 2*x - 2\).

Step 1: Compute the derivative of the function

The function \(f(x) = \frac{1}{2}x^2\), being a standard polynomial, can be differentiated using basic rules of derivatives. The derivative of an expression \(x^n\) where \(n\) is a constant is \(\frac{dx^n}{dx} = n*x^{n-1}\). Therefore, the derivative of the function is \(f'(x) = x\).

Step 2: Evaluate the derivative at the point

The slope of the tangent line at the point (2,2) is given by the derivative of the function at that point. Hence plug \(x = 2\) into \(f'(x)\). \(f'(2) = 2\). So the slope of the tangent line at the point (2,2) is 2.

Step 3: Formulate the equation of the tangent line

Now that we have the slope and coordinates of a point on the line, we can use the formula for the equation of the line which is \(y=mx+c\). Substituting \(m = 2\), and the coordinates of the point (2,2) into the formula gives \(y = 2*x - 2\). This is the equation of the tangent line.

Key Concepts

Tangent Line

In calculus, a tangent line is a straight line that just "touches" a curve at a particular point. This concept can be likened to touching a ball with a stick at one exact point
without cutting through the ball. At the point of contact, the slope of the tangent line is exactly the same as the slope of the curve. Finding the tangent line involves determining two main things:

• The point where the tangent line will touch the curve.
• The slope of the line at that specific point, which is equivalent to the derivative of the curve's equation at that point.

In simple terms, the tangent line offers a linear approximation of the curve near a specific point, showing how the function behaves in that tiny interval.

Derivative

The derivative plays a critical role in calculus—it is essentially the "rate of change" of a function. Think of it as how much the function value changes when you slightly change the input. In more practical terms, if you think of driving a car, the derivative of your position with respect to time is your speed. When dealing with polynomials like \(f(x) = \frac{1}{2}x^2\), the derivative indicates how steep or flat the function is at different points. The basic rule to derive polynomials is to bring down the power and reduce the power by one. For example, \(x^n\) becomes \(n*x^{n-1}\). For our specific function, applying the rule gives us \(f'(x) = x\). This derivative tells us that the slope of the curve increases linearly as we move along the x-axis. Evaluating \(f'(x)\) at \(x = 2\) gives us the slope of the tangent line at that point.

Polynomial

A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. These expressions are foundational in calculus because they are simple yet versatile forms of functions. The function \(f(x) = \frac{1}{2}x^2\) is a quadratic polynomial. It includes a single term with the variable \(x\) raised to the power of 2. Polynomials are smooth and continuous, which makes them very good candidates for studying with calculus tools like differentiation. Quadratic polynomials, like the one in this exercise, have parabolic shapes. Their derivative, being a linear function as seen in \(f'(x) = x\), makes it convenient to predict the function's rate of change and to construct tangent lines at specific points.

Limit Definition

The limit definition of a derivative is foundational to understanding how derivatives measure the rate of change. This concept was historically crucial, allowing us to find the instantaneous rate of change of a function at any point. The formal definition is: \[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\] This equation might look daunting, but it captures the idea of approximation: it observes how the function behaves as we take two infinitely close points on the function and shrink the distance between them to zero. This approach allows us to construct precise mathematical calculations of slopes and is crucial in finding the tangent line. Applying this to our function would involve finding this rate as \(h\) approaches zero, ultimately yielding the same derivative result as using polynomial rules: \(f'(x) = x\). This ensures we have a correct slope for our tangent line calculation.