Problem 9 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The derivative of \(y=\cos x\) is defined by $$ [\cos x]^{\prime}=\lim _{h \rightarrow 0} \frac{\cos (x+h)-\cos x}{h} $$ Make a plot of $$ y=-\sin t \quad \text { and of } \quad \frac{\cos (t+0.2)-\cos t}{0.2} \quad-\frac{\pi}{2} \leq t \leq 2 \pi $$ Repeat, using \(h=0.05\) instead of \(h=0.2\).

Step-by-Step Solution

Answer

Plot \( y = -\sin t \) and the approximation for \( h = 0.2 \) and \( h = 0.05 \), showing both approximations approach the derivative.

Step 1: Define the Functions

We need to define two functions for plotting. The first is the derivative, given as \( y = -\sin t \). The second is the approximation of the derivative using the limit definition, given by: \[\frac{\cos(t+h) - \cos(t)}{h}.\]We will start with \( h = 0.2 \), then later change \( h \) to 0.05.

Step 2: Create the t-values

Define the range of \( t \) over which to plot both functions. According to the problem, \( t \) varies from \(-\frac{\pi}{2}\) to \(2\pi\). Use a fine grid of points, like an array of 1000 evenly spaced points.

Step 3: Calculate Function Values for h=0.2

Calculate the values for each function at each \( t \) value. For \( y = -\sin t \), compute the sine for each \( t \). For the approximation, compute \(\frac{\cos(t+0.2) - \cos(t)}{0.2} \) for each \( t \).

Step 4: Plot the Functions for h=0.2

Plot both functions on the same graph. Use different colors or line styles to distinguish between \( y = -\sin t \) and \( \frac{\cos(t+0.2) - \cos(t)}{0.2} \). Set appropriate labels and legends.

Step 5: Calculate Function Values for h=0.05

Repeat Step 3, but this time set \( h = 0.05 \). Calculate the approximation of the derivative with the new \( h \) as:\[\frac{\cos(t+0.05) - \cos(t)}{0.05}.\]

Step 6: Plot the Functions for h=0.05

Plot \( y = -\sin t \) and \( \frac{\cos(t+0.05) - \cos(t)}{0.05} \) on the same graph, using different colors or line styles. Include labels and legends to identify which plot corresponds to which function.

Key Concepts

Limit Definition of Derivative

The derivative of a function gives us the rate at which the function is changing at any given point. The limit definition of the derivative for a function \(f(x)\) is expressed as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] This formula helps in finding the derivative by considering the value of the function at \(x + h\) and \(x\), and then dividing by \(h\), as \(h\) approaches zero. For the cosine function \(\cos x\), its derivative is \(-\sin x\). This derivation involves applying the limit definition with \(f(x) = \cos x\). By rewriting, we evaluate: \[ \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h} \] Applying trigonometric identities and limits rules, this becomes \(-\sin x\), which describes the rate of change of \(\cos x\).

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental in mathematics, especially in calculus. They describe periodic oscillations like waves and circular motions. In our exercise of finding the derivative of \(y = \cos x\), it's important to understand how these functions behave.

• The sine function, \(\sin x\), represents vertical oscillations on the unit circle.
• The cosine function, \(\cos x\), represents horizontal oscillations on the same circle.
• Both functions repeat values in intervals, making them periodic with a period of \(2\pi\).

When calculating the derivative using the limit definition, knowing these properties allows us to manipulate trigonometric identities. For example, understanding that \(\cos (x + h)\) can be expanded using angle addition formulas aids in simplifying the derivative's limit calculations.

Numerical Approximation

Numerical approximation plays a critical role when analytical solutions are complex or when dealing with continuous functions in practical applications. Instead of solving limits directly, we can approximate the derivative by choosing a small value for \(h\). This gives us the average rate of change over a small interval. In the exercise, the problem is computed for \(h = 0.2\) and \(h = 0.05\), estimating the derivative of \(\cos x\) as: \[ \frac{\cos(t+h) - \cos(t)}{h} \] A smaller \(h\) provides a closer approximation to the true derivative \(y = -\sin t\). In practical scenarios, numerical approximation is used in computers to calculate derivatives when limits cannot be determined analytically due to complexity or resource constraints.

Plotting Graphs

Visualizing functions through graphs is an excellent way to understand the behavior and properties of mathematical equations. In this exercise, plotting helps us compare the true derivative \(y = -\sin t\) with its numerical approximation: \[ \frac{\cos(t+0.2) - \cos(t)}{0.2} \] A plot across the range \(-\frac{\pi}{2} \leq t \leq 2\pi\) shows periodic cyclic patterns of trigonometric functions.

• The derivative \(y = -\sin t\) is graphed as a continuous line showing its oscillations.
• The approximation is depicted as dots or a separate line, highlighting any disparities with the sine graph.

As \(h\) becomes smaller, the graph of the approximation aligns closer with the derivative curve, revealing how numerical approximations improve with smaller intervals.