Problem 52 - University Calculus: Early Transcendentals

Question

Use a CAS to perform the following steps for the functions. 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). $$f(x)=\cos x+4 \sin (2 x), \quad x_{0}=\pi$$

Step-by-Step Solution

Answer

Plot the function, calculate the difference quotient, find its limit, and graph secant and tangent lines.

Step 1: Plot the Function

First, let's plot the function \(y = f(x) = \cos x + 4 \sin(2x)\) over the interval \( (x_0 - 1/2) \leq x \leq (x_0 + 3) \), with \( x_0 = \pi \). This means we plot over the interval \( (\pi - 1/2) \leq x \leq (\pi + 3) \). Use a CAS to generate the plot for a visual representation of the function.

Step 2: Define the Difference Quotient

The difference quotient is given by the expression \( q(h) = \frac{f(x_0 + h) - f(x_0)}{h} \). Substitute \( f(x) = \cos x + 4\sin(2x) \) and \( x_0 = \pi \) into this expression, which becomes \( q(h) = \frac{\cos(\pi + h) + 4\sin(2(\pi + h)) - (\cos \pi + 4\sin(2\pi))}{h} \). Enter this expression into your CAS to create a function of \( h \).

Step 3: Find the Limit of the Difference Quotient

Find the limit of the difference quotient \( q(h) \) as \( h \rightarrow 0 \). Use the CAS to evaluate \( \lim_{h \to 0} q(h) \). This limit gives the derivative \( f'(x) \) at \( x = x_0 = \pi \).

Step 4: Define Secant Lines and Graph Together

Define the secant lines using the formula \( y = f(x_0) + q(h) \cdot (x - x_0) \) for \( h = 3, 2, 1 \). For each value of \( h \), substitute into the equation and graph the lines on the same plot as \( f(x) \). Also, include the tangent line found from \( \lim_{h \to 0} q(h) \) \((f'(x_0))\) over the interval. Use CAS to combine these graphs visually.

Key Concepts

Difference Quotient

The difference quotient is a crucial element in calculus used to understand how functions change. It represents the average rate of change of a function over a small interval \(h\). Think of \(q(h) = \frac{f(x_0 + h) - f(x_0)}{h}\) as a formula that measures how much \(f(x)\) changes from \(x_0\) to \(x_0 + h\).

Here, \(f(x) = \cos(x) + 4\sin(2x)\) when \(x_0 = \pi\), so you substitute this into the difference quotient to see how the function behaves around \(x = \pi\).

- By entering this into a Computer Algebra System (CAS), you create a new function of \(h\), which expresses change as \(h\) varies.- It’s about exploring the concept of derivatives even before limits come into play.- The smaller the value of \(h\), the closer \(q(h)\) approximates the true rate of change at a point.

Secant Line

The secant line connects two points on the graph of a function, imagining that line between \(x_0\) and \(x_0 + h\). It's a straight line that offers a snapshot of the average rate of change over that interval. This line is expressed as \(y = f(x_0) + q(h) \cdot (x - x_0)\).

For our function \(f(x) = \cos(x) + 4\sin(2x)\) at \(x_0 = \pi\), you calculate this for various \(h\) (like 3, 2, and 1).

- Each of these secant lines shows a different average rate of change within those intervals.- By graphing these secant lines together with the function, you visualize how the line interacts with \(f(x)\).- A smaller \(h\) narrows the interval and makes the secant line approach the tangent line.

Tangent Line

The tangent line is the next leap from the secant line. Imagine shrinking that line segment between \(x_0\) and \(x_0 + h\) down to a point where \(h\) approaches zero. The line that results is the tangent line, representing the instantaneous rate of change, or the derivative, at \(x_0\).

Calculating the limit of the difference quotient \(q(h)\) as \(h \rightarrow 0\) is how you find the slope of this tangent line.

- When \(h\) is exactly zero, you are left with the tangent line slope — the true derivative at a point.- This tangent is particularly useful because it helps predict how the function behaves very close to \(x_0\), a major player in calculus.- It provides a clear visual and numerical insight on how the function changes instantaneously.

Limit

In calculus, the concept of a limit helps you understand what happens as a value approaches another value. For the difference quotient \(q(h)\), finding the limit as \(h\) approaches zero is key to finding the tangent line.

So how does it work?- For \(q(h) = \frac{f(x_0 + h) - f(x_0)}{h}\), you calculate the limit \(\lim_{h \to 0} q(h)\).- This gives the exact slope of the tangent line, providing the true derivative \(f'(x_0)\) at that point.

Grasping limits is a central part of calculus, as it allows you to understand how functions behave at specific points by approaching but never actually reaching the target value.

Graphing Functions

Graphing is an indispensable tool in calculus, making abstract concepts concrete. You can visually grasp function behaviors, lines, and their intersections. For instance, plotting \(f(x) = \cos(x) + 4\sin(2x)\) over a given interval helps visualize where and how it changes.

- When you overlay secant and tangent lines on the graph, it highlights differences between average changes and instantaneous changes.- Use a Computer Algebra System (CAS) to generate these plots effortlessly for accurate visual representation.- Graphing secant lines for various \(h\) values alongside the tangent line lets you see convergence visually, offering an intuitive understanding of calculus concepts.