Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) What is the limit definition of \(\left.\frac{d}{d x} \cos x\right|_{x=0}\) ? (b) Numerically approximate \(\left.\frac{d}{d x} \cos x\right|_{x=0}\)

Step-by-Step Solution

Answer

The calculated value of the derivative of \( \cos x \) at \( x=0 \) is 0 and the numerical approximation with \( h=0.0001 \) should also be very close to 0.

Step 1: Understand the limit definition of the derivative

The limit definition of the derivative of a function at a point x is given by: \( f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \). Let's apply this definition to our function, \( f(x) = \cos x \).

Step 2: Substituting the function into the definition

Substituting the function \( f(x) = \cos x \) into the limit definition of the derivative, we get \( f'(x) = \lim_{h \rightarrow 0} \frac{\cos(x+h) - \cos x}{h} \)

Step 3: Evaluating the derivative at x=0

Since we want to evaluate the derivative at \( x=0 \), we substitute \( x=0 \) into the limit, giving us: \( f'(0) = \lim_{h \rightarrow 0} \frac{\cos(0+h) - \cos 0}{h} = \lim_{h \rightarrow 0} \frac{\cos h - 1}{h} \)

Step 4: Using trigonometric identities and limits

By using trigonometric identities, it's known that \( \lim_{h \rightarrow 0} \frac{\cos h - 1}{h} = 0 \) which implies \( f'(0) = 0 \)

Step 5: Numerical approximation of the derivative

Numerical approximation of the derivative can be done using the same limit definition of the derivative but keeping \( h \) to be a small number. For a relatively small h, let's say 0.0001, the numerical approximation will be close to the actual value: \( f'(0) \approx \frac{\cos(0+0.0001) - \cos 0}{0.0001} \)

Key Concepts

Numerical Approximation

Numerical approximation is a valuable method when working with derivatives, especially when exact symbolic manipulation is complex or impractical. For many functions, or in specific scenarios, calculating a derivative exactly using limits and symbols might be challenging. This is where numerical approximation comes in handy. By choosing a very small value for the variable increment, often denoted by the letter \( h \), we can approximate the derivative's value quite accurately. In this exercise, we use a tiny \( h \) such as 0.0001 to estimate the derivative of the function \( \cos x \) at \( x = 0 \). The formula used is:

• \( f'(0) \approx \frac{\cos(0+0.0001) - \cos 0}{0.0001} \)

This calculation will give you a numerical result very close to zero, aligning well with the exact derivative's value at this point. This approach provides a great check to confirm your understanding of the derivative process by comparing it to algebraic findings.

Trigonometric Identities

Trigonometric identities are crucial tools in calculus, particularly when differentiating trigonometric functions. These identities simplify expressions and help find exact values for limits involving trigonometric functions. In the context of derivatives, especially when applying the limit definition, understanding and utilizing these identities can significantly ease the process. Consider the limit expression:

• \( \lim_{h \rightarrow 0} \frac{\cos h - 1}{h} = 0 \)

This specific identity is essential in this exercise. Knowing this identity allows you to determine quickly that the derivative of \( \cos x \) at \( x = 0 \) is zero. Easy access to, and familiarity with, trigonometric identities can be a major asset when performing differentiation or integration of trigonometric functions.

Differentiation of Trigonometric Functions

Differentiating trigonometric functions is a fundamental skill in calculus, involving the calculation of derivatives for functions such as sine, cosine, and tangent. These derivatives are key in understanding the rate at which trigonometric functions change. When using the limit definition of derivatives, the knowledge of these derivatives aids in simplifying complex problems and distinguishing patterns in function behavior.For \( \cos x \), the derivative can be found using the limit definition:

• \( f'(x) = \lim_{h \rightarrow 0} \frac{\cos(x+h) - \cos x}{h} \)

By applying known trigonometric limits and identities, it becomes clear that this simplifies to zero when \( x = 0 \). The process of differentiation unveils that tangent slope at specific points, providing insights into how the function behaves, which is pivotal for broader applications across different disciplines.