Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

For small positive values of \(z, \sin z

Step-by-Step Solution

Answer

For small z, \( z - \sin z \) is tiny and \( \frac{\sin z}{z} \approx 1 \); the derivative at \( t=0 \) is 1.

Step 1: Calculate z - sin z for z=0.1

First, calculate \( \sin z \) for \( z = 0.1 \). Using a calculator, \( \sin 0.1 \approx 0.09983 \). Now calculate \( z - \sin z = 0.1 - 0.09983 = 0.00017 \).

Step 2: Calculate z - sin z for z=0.01

Next, calculate \( \sin z \) for \( z = 0.01 \). Using a calculator, \( \sin 0.01 \approx 0.00999983 \). Now calculate \( z - \sin z = 0.01 - 0.00999983 = 0.00000017 \).

Step 3: Calculate z - sin z for z=0.001

Now, calculate \( \sin z \) for \( z = 0.001 \). Using a calculator, \( \sin 0.001 \approx 0.0009999998 \). Calculate \( z - \sin z = 0.001 - 0.0009999998 = 0.0000000002 \).

Step 4: Calculate (sin z)/z for z=0.1

To find \( \frac{\sin z}{z} \) for \( z = 0.1 \), use \( \sin 0.1 \approx 0.09983 \). So, \( \frac{\sin 0.1}{0.1} \approx 0.9983 \).

Step 5: Calculate (sin z)/z for z=0.01

For \( z = 0.01 \), use \( \sin 0.01 \approx 0.00999983 \). Hence, \( \frac{\sin 0.01}{0.01} \approx 0.999983 \).

Step 6: Calculate (sin z)/z for z=0.001

For \( z = 0.001 \), \( \sin 0.001 \approx 0.0009999998 \). Thus, \( \frac{\sin 0.001}{0.001} \approx 0.9999998 \).

Step 7: Estimate the derivative of sin t at t=0

From calculus, the limit \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \). Thus, the best estimate of the derivative \( [\sin t]_{t=0}^{'} \) is \( 1 \).

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In mathematical terms:

• In a right triangle: \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

For small values of \( z \), it can be observed that \( \sin z \approx z \). This is because as \( z \rightarrow 0 \), \( \sin z \) approaches \( z \).

In calculus, the behavior of the sine function for small angles is crucial. You can see from exercise part (a) that \( z - \sin z \) gives us a tiny positive value that gets smaller as \( z \) approaches zero, which illustrates how close \( \sin z \) is to \( z \).

This approximation becomes very significant in various mathematical and physical applications, like measuring small oscillations or angles.

Derivatives

A derivative of a function at a point provides the slope of the tangent line at that point. It's a measure of how a function changes as its input changes.

The derivative of the sine function, \( \sin t \), at zero illustrates a remarkable and important property:

• The derivative \([ \sin t]_{t=0}^{'}\) is evaluated using the limit \( \lim_{h \rightarrow 0} \frac{\sin h}{h} \).
• This limit evaluates to 1, indicating that the slope of the sine function at \( t=0 \) is 1.

This tells us that near zero, \( \sin t \) increases linearly with a slope of 1 as \( t \) increases.

Understanding this concept helps in grasping fundamental concepts of calculus such as instantaneous rate of change, and also plays a vital role in solving more complex problems.

Limits

The concept of a limit is a fundamental building block in calculus, representing the value that a function or sequence 'approaches' as the input approaches some value.

In the context of this exercise, the expression \( \lim_{h \rightarrow 0} \frac{\sin h}{h} \) is key. It shows how limits help to find derivatives, especially at points where they seem not directly computable.

• This specific limit equals 1, which is essential for establishing the derivative of \( \sin t \) at \( t=0 \).
• Using limits to calculate derivatives helps provide insights into the behavior of functions at specific points, particularly at those where direct substitution is difficult or yields indeterminate forms.

Mastering limits and their applications forms the foundation needed to tackle more advanced topics in calculus and aids in understanding the continuity and differentiability of functions across different fields in mathematics and engineering.