[Technology Exercise]


Graph y = 1/(2√x) in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graph
y = (√(x + h) − √x)/h
for h = 1, 0.5, 0.1. Then try h = −1, −0.5, −0.1. Explain what is going on.

Verified step by step guidance

1. Step 1: Begin by understanding the function y = 1/(2√x). This is a transformation of the basic square root function. The graph of y = √x is a curve that starts at the origin (0,0) and increases slowly. The function y = 1/(2√x) is a reciprocal function, which means it will have a vertical asymptote at x = 0 and will decrease as x increases.
2. Step 2: Set up your graphing window to display the range 0 ≤ x ≤ 2. This will allow you to see the behavior of the function y = 1/(2√x) within this interval. The graph will start from a high value at x = 0 and decrease as x approaches 2.
3. Step 3: Next, graph the function y = (√(x + h) − √x)/h for different values of h. Start with h = 1. This function represents the difference quotient, which is used to approximate the derivative of √x. As h approaches 0, this function approximates the slope of the tangent line to the curve y = √x at a given point.
4. Step 4: Repeat the graphing for h = 0.5 and h = 0.1. As h gets smaller, the graph of y = (√(x + h) − √x)/h should get closer to the derivative of y = √x, which is 1/(2√x). This is because the difference quotient becomes a better approximation of the derivative as h approaches 0.
5. Step 5: Now, try negative values for h: h = -1, -0.5, -0.1. When h is negative, the function y = (√(x + h) − √x)/h represents the slope of the secant line from x to x + h, where x + h is less than x. This will give you a negative slope, and as h approaches 0 from the negative side, the graph should again approach the derivative of y = √x, which is 1/(2√x). This demonstrates the concept of the derivative from both the left and right sides.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of mathematical expressions. For y = 1/(2√x), understanding how the function behaves as x approaches 0 and 2 is crucial. This function is a transformation of the basic square root function, scaled and inverted, which affects its shape and position on the graph.

Limits and Continuity

Limits help us understand the behavior of functions as they approach specific points or infinity. The expression y = (√(x + h) − √x)/h is a difference quotient, which is used to approximate the derivative of √x. As h approaches zero, this quotient gives insight into the instantaneous rate of change, highlighting the concept of limits and continuity in calculus.

Derivative Approximation

The derivative of a function represents its rate of change. The expression y = (√(x + h) − √x)/h approximates the derivative of √x using finite differences. By varying h, we observe how the approximation improves as h approaches zero, demonstrating the fundamental idea of derivatives as limits of difference quotients, essential for understanding calculus.