Problem 36 - Calculus Early Transcendentals

Question

Use a graphing utility to graph the following on the same screen: the curve \(y=x^{2} / 4,\) the tangent line to this curve at \(x=1,\) and the secant line joining the points \((0,0)\) and \((2,1)\) on this curve.

Step-by-Step Solution

Answer

Graph the equations: \(y = \frac{x^2}{4}\), \(y = \frac{1}{2}x - \frac{1}{4}\), and \(y = \frac{1}{2}x\) on the same screen.

Step 1: Define the Curve Equation

The given curve is defined by the equation \(y = \frac{x^2}{4}\). This is a parabola that opens upwards.

Step 2: Find the Tangent Line at x=1

To find the tangent line to the curve at \(x = 1\), first calculate the derivative of the function \(y = \frac{x^2}{4}\). The derivative \(y'\) represents the slope of the tangent line at any point. The derivative is \(y' = \frac{d}{dx} \left( \frac{x^2}{4} \right) = \frac{1}{2}x\). At \(x = 1\), the slope \(m\) is \(\frac{1}{2}\). Plug \(x = 1\) into the original function to find the point of tangency: \(y = \frac{1^2}{4} = \frac{1}{4}\), giving the point \((1, \frac{1}{4})\). The equation of the tangent line is then \(y - \frac{1}{4} = \frac{1}{2}(x - 1)\). Simplifying, \(y = \frac{1}{2}x - \frac{1}{4}\).

Step 3: Find the Secant Line Equation

The secant line joins two points on the curve, \((0,0)\) and \((2,1)\). The slope of the secant line is \(m = \frac{1 - 0}{2 - 0} = \frac{1}{2}\). Using the point-slope form with point \((0,0)\), the equation is \(y - 0 = \frac{1}{2}(x - 0)\), or simply \(y = \frac{1}{2}x\).

Step 4: Graph the Equations

On a graphing utility, plot the curve \(y = \frac{x^2}{4}\), the tangent line \(y = \frac{1}{2}x - \frac{1}{4}\), and the secant line \(y = \frac{1}{2}x\) on the same screen. This will show the parabola, and how the tangent and secant lines compare visually at the point of tangency and between the two points respectively.

Key Concepts

Tangent Line

When we talk about a tangent line in calculus, we are referring to a straight line that just "touches" a curve at one single point. It doesn't cut through the curve, it just grazes it at exactly that spot. This line represents the slope or gradient of the curve at that particular point.

To find this tangent line, we first take the derivative of the curve’s equation. In this example, the curve is defined by the equation \(y = \frac{x^2}{4}\), which is a parabola. Calculating the derivative \(y' = \frac{1}{2}x\), we determine the slope of the tangent at any point \(x\) on the curve. At \(x = 1\), this gives us a slope of \(\frac{1}{2}\).

• At \(x = 1\), the curve passes through the point \((1, \frac{1}{4})\).
• Using this point and the slope, the tangent line is \(y = \frac{1}{2}x - \frac{1}{4}\).

The tangent line helps visualize how the slope of a curve behaves locally around a point. It can tell you how steep the curve is at that very spot.

Secant Line

A secant line, unlike a tangent line, intersects a curve at two or more points. In our exercise, the secant line connects two specific points on the curve: \((0,0)\) and \((2,1)\).

The secant line gives a broader perspective on the average rate of change between these two points. Essentially, its slope indicates how much the function rises over the distance it covers between these points. Calculating the slope \(m\) gives us \(\frac{1 - 0}{2 - 0} = \frac{1}{2}\).

• Using the slope and the point \((0,0)\), the equation of the secant line simplifies to \(y = \frac{1}{2}x\).

Graphing this line shows the average increase of the parabola from point \((0,0)\) to \((2,1)\), which is useful for understanding how the function behaves between these two locations.

Parabola

A parabola is a symmetric curve that can open upwards or downwards. For this exercise, we are dealing with the equation \(y = \frac{x^2}{4}\), which creates a parabola that opens upwards. This specific shape is due to the squared term in the equation.

Parabolas have key features:

• The vertex, which is the highest or lowest point of the curve, for \(y = \frac{x^2}{4}\) is at the origin \((0,0)\).
• Since the equation is \(\frac{x^2}{4}\), it makes the parabola wider compared to \(x^2\). The fraction \(\frac{1}{4}\) is a vertical stretch/compression factor.

To graph this parabola, you can plot some points by choosing x-values, plugging them into the equation, and finding the corresponding y-values. This will show the curve's characteristic shape, and help to visualize how the tangent and secant lines relate to it, allowing for a deeper understanding of its overall geometry.