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). $$ f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2 $$

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).
$$
f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2
$$

Key Concepts

Graphing

Graphing is the process of plotting functions or equations on a coordinate system to visually analyze their behavior. It is vital in understanding relationships, trends, and intersections between different mathematical entities, such as functions and their tangent or secant lines.

Step Size (h)

The step size, denoted by h, is the small increment in the variable used in the difference quotient. It is a critical parameter in numerical approximations of derivatives, and as it approaches zero, the difference quotient provides a more accurate estimation of the instantaneous rate of change.

Tangent Line

A tangent line to a curve at a given point is a line that touches the curve at that point and has the same slope as the curve there. It represents the instantaneous rate of change of the function and provides a linear approximation of the function near the point of tangency.

Secant Line

A secant line is a straight line intersecting a function's graph at two distinct points. It provides an approximation of the function's slope over an interval. As the interval shrinks, the secant line approaches being a tangent line, thereby better approximating the derivative at a point.

Limit

A limit describes the behavior of a function as its input approaches a particular value. It is crucial in calculus for defining concepts such as continuity and the derivative. In the context of the difference quotient, taking the limit as the step size approaches zero yields the derivative, representing the instantaneous rate of change.

Difference Quotient

The difference quotient is a fundamental concept used to approximate the derivative of a function. It measures the average rate of change of the function over a small interval and is expressed as the ratio of the change in function value to the change in the variable. This concept underpins the formal definition of the derivative.

Computer Algebra System (CAS)

A Computer Algebra System is a software tool that facilitates symbolic mathematics, including algebraic manipulations, solving equations, and plotting functions. It is essential for exploring mathematical problems, visualizing functions, and performing complex computations that would be tedious by hand.

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