Let's break down the assignment step by step for **each function** \( f(x), g(x), k(x) \). --- ## **1. Compute the Derivative Analytically** ### **Function 1: \( f(x) = x^2 + 2x - 1 \)** #### **Step 1: Compute the Derivative** \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] \[ f(x+h) = (x+h)^2 + 2(x+h) - 1 = x^2 + 2xh + h^2 + 2x + 2h - 1 \] \[ f(x+h) - f(x) = (x^2 + 2xh + h^2 + 2x + 2h - 1) - (x^2 + 2x - 1) = 2xh + h^2 + 2h \] \[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 2h}{h} = 2x + h + 2 \] \[ \lim_{h \to 0} (2x + h + 2) = 2x + 2 \] \[ \boxed{f'(x) = 2x + 2} \] --- ### **Function 2: \( g(x) = \frac{-500}{x}, \; x \neq 0 \)** #### **Step 1: Compute the Derivative** \[ g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} \] \[ g(x+h) = \frac{-500}{x+h} \] \[ \frac{g(x+h) - g(x)}{h} = \frac{\frac{-500}{x+h} - \frac{-500}{x}}{h} = \frac{-500 \left(\frac{1}{x+h} - \frac{1}{x}\right)}{h} = \frac{-500 \left(\frac{x - (x+h)}{x(x+h)}\right)}{h} = \frac{-500 \left(\frac{-h}{x(x+h)}\right)}{h} = \frac{500h}{h x(x+h)} = \frac{500}{x(x+h)} \] \[ \lim_{h \to 0} \frac{500}{x(x+h)} = \frac{500}{x^2} \] \[ \boxed{g'(x) = \frac{500}{x^2}} \] --- ### **Function 3: \( k(x) = |x - 2| \)** #### **Step 1: Compute the Derivative (Piecewise)** - For \( x > 2: \quad k(x) = x - 2 \rightarrow k'(x) = 1 \) - For \( x < 2: \quad k(x) = -(x - 2) = -x + 2 \rightarrow k'(x) = -1 \) - At \( x = 2 \): The left and right derivatives are \( -1 \) and \( 1 \), so the derivative **does not exist** at \( x = 2 \). \[ k'(x) = \begin{cases} 1, & x > 2 \\ -1, & x < 2 \\ \text{undefined}, & x = 2 \end{cases} \] --- ## **2. Graph the Derivative** - On your calculator, enter: - \( Y_1 = 2X + 2 \) - \( Y_1 = \frac{500}{X^2} \) (restrict domain to \( X \neq 0 \)) - \( Y_1 = \text{piecewise: 1 for } X > 2, -1 \text{ for } X < 2 \) (no value at \( X = 2 \)) ## **3. Graph the Numerical Derivative** - Use the calculator's `nDeriv` function: - \( Y_2 = nDeriv(f(X), X, X) \) - \( Y_2 = nDeriv(g(X), X, X) \) - \( Y_2 = nDeriv(k(X), X, X) \) ## **4. Compare and Discuss** - Plot \( Y_1 \) and \( Y_2 \) together. - Set \( Y_3 = Y_2 - Y_1 \) and look for differences. - Discuss where the graphs match or differ (e.g., at \( x = 2 \) for \( k(x) \), \( x = 0 \) for \( g(x) \)), and how accurate the numerical derivative is. --- ## **Summary Table** | Function | Analytic Derivative | Notable Points | |------------|----------------------------------|-------------------------------| | \( f(x) \) | \( 2x + 2 \) | Linear; matches exactly | | \( g(x) \) | \( \frac{500}{x^2} \) | Undefined at \( x = 0 \) | | \( k(x) \) | \( 1 \) if \( x>2 \), \( -1 \) if \( x<2 \); undefined at \( x=2 \) | Sharp corner at \( x=2 \) | --- ### **Final Comment Example** > The analytic and numerical derivatives matched very closely except at points where the derivative is undefined or changes abruptly (like at \( x = 2 \) for \( k(x) \)). The `nDeriv` function on the calculator approximates the derivative well except near discontinuities or cusps, where it may not reflect the undefined nature of the derivative. --- **If you need step-by-step calculator keystrokes, let me know which calculator model you are using!**

Certainly! Let's go through each function step-by-step, showing detailed calculations, and then summarize the final answers. --- ### **Function 1: \( f(x) = x^2 + 2x - 1 \)** #### **Step 1: Compute the derivative \( f'(x) \) using the limit definition** \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Calculate \( f(x+h) \): \[ f(x+h) = (x+h)^2 + 2(x+h) - 1 \] Expand: \[ (x+h)^2 = x^2 + 2xh + h^2 \] \[ 2(x+h) = 2x + 2h \] So, \[ f(x+h) = x^2 + 2xh + h^2 + 2x + 2h - 1 \] Now, subtract \( f(x) = x^2 + 2x - 1 \): \[ f(x+h) - f(x) = (x^2 + 2xh + h^2 + 2x + 2h - 1) - (x^2 + 2x - 1) \] Simplify: \[ = 2xh + h^2 + 2h \] Divide by \( h \): \[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 2h}{h} = 2x + h + 2 \] Take the limit as \( h \to 0 \): \[ f'(x) = \lim_{h \to 0} (2x + h + 2) = 2x + 2 \] **Final derivative:** \[ \boxed{f'(x) = 2x + 2} \] --- ### **Function 2: \( g(x) = -\frac{500}{x} \), \( x \neq 0 \)** #### **Step 1: Compute the derivative \( g'(x) \) using the limit definition** \[ g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} \] Calculate \( g(x+h) \): \[ g(x+h) = -\frac{500}{x+h} \] Subtract: \[ g(x+h) - g(x) = -\frac{500}{x+h} - \left(-\frac{500}{x}\right) = -\frac{500}{x+h} + \frac{500}{x} \] Combine over a common denominator: \[ = \frac{500}{x} - \frac{500}{x+h} = \frac{500(x+h) - 500x}{x(x+h)} = \frac{500x + 500h - 500x}{x(x+h)} = \frac{500h}{x(x+h)} \] Divide by \( h \): \[ \frac{g(x+h) - g(x)}{h} = \frac{500h}{x(x+h) \cdot h} = \frac{500}{x(x+h)} \] Now, take the limit as \( h \to 0 \): \[ g'(x) = \lim_{h \to 0} \frac{500}{x(x+h)} = \frac{500}{x \cdot x} = \frac{500}{x^2} \] **Final derivative:** \[ \boxed{g'(x) = \frac{500}{x^2}} \] --- ### **Function 3: \( k(x) = |x - 2| \)** This function involves an absolute value, so the derivative depends on the sign of \( x - 2 \). #### **Step 1: Understand the piecewise nature** - For \( x > 2 \): \[ k(x) = x - 2 \quad \Rightarrow \quad k'(x) = 1 \] - For \( x < 2 \): \[ k(x) = -(x - 2) = -x + 2 \quad \Rightarrow \quad k'(x) = -1 \] - At \( x = 2 \): \[ k(x) \text{ has a cusp; derivative does not exist at } x=2 \] **Final piecewise derivative:** \[ k'(x) = \begin{cases} 1, & x > 2 \\ -1, & x < 2 \\ \text{undefined at } x=2 \end{cases} \] --- ## **Summary of derivatives:** | Function | Derivative \( f'(x) \) | Explanation | |---|---|---| | \( f(x) = x^2 + 2x - 1 \) | \( 2x + 2 \) | Polynomial, straightforward derivative | | \( g(x) = -500/x \) | \( \frac{500}{x^2} \) | Power rule with negative exponent | | \( k(x) = |x - 2| \) | piecewise \( 1 \) (for \( x>2 \)), \(-1\) (for \( x<2 \)), undefined at 2 | Absolute value function | --- ## **Next Steps:** - **Graph each derivative**: - For \( f'(x) = 2x + 2 \), graph a straight line. - For \( g'(x) = 500/x^2 \), graph a positive curve approaching infinity near \( x=0 \). - For \( k'(x) \), plot a step function with values \( 1 \) and \( -1 \), undefined at \( x=2 \). - **Graph the numerical derivative**: - Use the calculator's `nDeriv` function with the original functions. - Compare the numerical results with the analytical derivatives. - **Analysis**: - Look for differences, especially near points where derivatives are undefined or change abruptly. - Discuss the accuracy of `nDeriv` at various points. --- ### **Final comment example:** > The numerical derivatives closely approximate the analytical derivatives where the functions are smooth and continuous. However, near points of discontinuity or where the derivative is undefined (e.g., at \( x=2 \) for \( k(x) \)), the `nDeriv` function may produce less accurate or undefined results. This illustrates that numerical methods are most reliable in regions where the function is smooth and well-behaved. --- **Please specify your calculator model if you need exact keysteps for inputting these functions!**