COMPUTER EXPLORATIONS Use a CAS to perform the following...

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises $47-50$ : a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$ b. Holding $x_{0}$ fixed, the difference quotient $$ q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h} $$ at $x_{0}$ becomes a function of the step size $h$ . Enter this function into your CAS workspace. c. Find the limit of $q$ as $h \rightarrow 0$ . d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ , and $1 .$ Graph them together with $f$ and the tangent line over the interval in part (a). $$ f(x)=x+\frac{5}{x}, \quad x_{0}=1 $$

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises $47-50$ :
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h$ . Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$ .
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ , and $1 .$ Graph them together with $f$ and the tangent line over the interval in part (a).
$$
f(x)=x+\frac{5}{x}, \quad x_{0}=1
$$

Key Concepts

Computer Algebra System (CAS) Usage

A Computer Algebra System (CAS) is a software tool that enables symbolic manipulation of mathematical expressions, graphing functions, and performing complex calculations. It is useful for exploring mathematical concepts interactively, including plotting functions and automating the process of differentiation and limit evaluation.

Function Plotting

Plotting a function involves graphing the equation on a coordinate system to visually analyze its behavior over a specified interval. This technique helps in understanding the overall shape of the function, identifying key features such as intercepts, asymptotes, and local extrema, and providing a visual context for subsequent analytical tasks.

Difference Quotient

The difference quotient is an expression that calculates the average rate of change of a function over a small interval. It is defined as [f(x0 + h) - f(x0)]/h, where h represents the change in the input. This concept is fundamental in understanding how small changes in the input translate into changes in the output and serves as the basis for the definition of the derivative.

Limit of the Difference Quotient

Taking the limit of the difference quotient as h approaches zero transitions the average rate of change into the exact instantaneous rate of change at a point. This process essentially leads to the concept of the derivative, which is a central idea in differential calculus for analyzing how functions change.

Secant Lines and Tangent Lines

A secant line is a line that intersects a curve at two or more points, providing an approximation of the function's slope over an interval. As the interval shrinks, the secant line approaches the tangent line, which touches the curve at exactly one point and represents the instantaneous rate of change at that point. The tangent line is used to approximate the behavior of the function near that point.

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