Questions: Δy/Δx = (f(x₂) - f(x₁))/(x₂ - x₁) = (f(x₁ + h) - f(x₁))/h, h ≠ 0

Δy/Δx = (f(x₂) - f(x₁))/(x₂ - x₁) = (f(x₁ + h) - f(x₁))/h, h ≠ 0
Transcript text: $\frac{\Delta y}{\Delta x}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}=\frac{f\left(x_{1}+h\right)-f\left(x_{1}\right)}{h}, \quad h \neq 0$
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Solution

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Solution Steps

To solve the given problem, we need to compute the difference quotient, which is a fundamental concept in calculus for finding the slope of the secant line between two points on the function \( f(x) \). The difference quotient is given by:

\[ \frac{f(x_1 + h) - f(x_1)}{h} \]

where \( h \neq 0 \). We will write a Python function to compute this difference quotient for a given function \( f \), a point \( x_1 \), and a small value of \( h \).

Step 1: Define the Difference Quotient

The difference quotient is given by:

\[ \frac{f(x_1 + h) - f(x_1)}{h} \]

where \( h \neq 0 \).

Step 2: Define the Function

Consider the function \( f(x) = x^2 \).

Step 3: Compute the Difference Quotient

Given:

  • \( x_1 = 1 \)
  • \( h = 0.01 \)

We compute:

\[ f(x_1 + h) = f(1 + 0.01) = f(1.01) = (1.01)^2 = 1.0201 \]

\[ f(x_1) = f(1) = 1^2 = 1 \]

\[ \frac{f(1.01) - f(1)}{0.01} = \frac{1.0201 - 1}{0.01} = \frac{0.0201}{0.01} = 2.01 \]

Final Answer

\[ \boxed{\frac{f(x_1 + h) - f(x_1)}{h} = 2.01} \]

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