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Chapter 3: Problem 59

By computing the first few derivatives and looking for a pattern, find \(d^{999} / d x^{999}(\cos x)\)

Short Answer

Expert verified
The 999th derivative of \( \cos x \) is \( \sin x \).

Step by step solution

01

Understand the Problem

We are asked to find the 999th derivative of the function \( \cos x \). The question implies a repetitive pattern in the higher-order derivatives, which we'll discover by computing the first few derivatives.
02

Compute the First Derivative

The first derivative of \( \cos x \) is \( d/dx(\cos x) = -\sin x \). We mark this as \( f^{(1)}(x) = -\sin x \).
03

Compute the Second Derivative

The second derivative is found by differentiating \( -\sin x \):\( d/dx(-\sin x) = -\cos x \). Mark this as \( f^{(2)}(x) = -\cos x \).
04

Compute the Third Derivative

The third derivative is obtained by differentiating \( -\cos x \): \( d/dx(-\cos x) = \sin x \). We label this as \( f^{(3)}(x) = \sin x \).
05

Compute the Fourth Derivative

Differentiate \( \sin x \) to get \( d/dx(\sin x) = \cos x \). Identify this as \( f^{(4)}(x) = \cos x \).
06

Identify the Pattern

From the derivatives computed: \( f^{(1)}(x) = -\sin x \), \( f^{(2)}(x) = -\cos x \), \( f^{(3)}(x) = \sin x \), and \( f^{(4)}(x) = \cos x \), the derivatives repeat every 4 steps: \( \cos x, -\sin x, -\cos x, \sin x \).
07

Calculate 999th Derivative Using the Pattern

Determine which function the 999th derivative corresponds to using modulo division: \( 999 \mod 4 = 3 \). Hence, \( f^{(999)}(x) = f^{(3)}(x) = \sin x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, denoted by \( \cos x \), is one of the fundamental trigonometric functions representing the x-coordinate of a point on the unit circle. It frequently shows up in various fields ranging from physics to engineering.
  • Cosine function is periodic, meaning it repeats its values at regular intervals. Specifically, \( \cos x \) has a period of \( 2\pi \), implying \( \cos(x + 2\pi) = \cos x \).
  • The graph of \( \cos x \) is a wave-like pattern, starting at its maximum (1), slowly decreasing to a minimum (-1), and returning to the maximum in one cycle.
  • The symmetry of \( \cos x \) is even, suggesting \( \cos(-x) = \cos x \). This property is particularly useful when dealing with derivatives, simplifying the calculations.
The cosine function's cyclical behavior is the key reason why its derivatives repeat in a specific pattern, making it easier to compute higher-order derivatives efficiently.
Derivative Pattern
The elegance of calculus shows itself in how functions behave under differentiation, especially periodic functions like the cosine. When differentiating \( \cos x \), you'll see that there is a cyclic pattern in its derivatives. To understand why this happens, let's look at the progression.
  • **First derivative:** \( \frac{d}{dx}(\cos x) = -\sin x \). Here, the cosine turns into sine, showcasing the intertwined nature of sine and cosine functions.
  • **Second derivative:** Differentiating \( -\sin x \) gives \( \frac{d}{dx}(-\sin x) = -\cos x \).
  • **Third derivative:** Continuing with \( -\cos x \) results in \( \sin x \).
  • **Fourth derivative:** Finally, differentiating \( \sin x \) gives \( \cos x \), completing the cycle back to the original function.
Each differentiation cycle, comprised of four stages, returns to the original function, allowing us to predict any higher-order derivative by recognizing this four-step repeat cycle. Thus, the 999th derivative corresponds to the third function in the sequence due to \( 999 \mod 4 = 3 \).
Trigonometric Differentiation
In calculus, trigonometric functions and their derivatives serve as an adventure playground for learning patterns. Differentiating trig functions involves some unique aspects due to their periodic nature.
  • **Sine and Cosine:** For these functions, the differentiation process results in sine turning into cosine and vice versa. Importously, with each derivative, there is a change in sign, emphasizing the close relationship between these functions.
  • **Periodicity:** Both sine and cosine repeat every \( 2\pi \), which simplifies this type of differentiation compared to polynomial functions, which do not naturally cycle.
By taking these derivatives, you further explore the symmetries and periodicities inherent in these functions. It's important to remember that even with higher derivatives, the process of differentiation reveals straightforward cyclic patterns seen within the sine and cosine functions.
Higher-Order Derivatives
Higher-order derivatives are derivatives of derivatives, extending to the nth derivative of a function. When dealing with these, especially in trigonometric functions, recognizing patterns significantly simplifies the task.
  • **Cosine and its derivatives:** We know that after four derivatives, cosine comes back to itself. Therefore, any integer-order derivative can easily be reduced to the remainder when divided by four.
  • **Simplification through patterns:** For the 999th derivative of \( \cos x \), you compute this using the pattern repeat length of four: \( 999 \mod 4 = 3 \). This indicates that the 999th derivative is equivalent to the third in the cycle, \( \sin x \).
  • **Applications:** Understanding these patterns helps in fields like signal processing or analyzing waves, where calculations involve understanding repetitive cycles.
Higher derivatives in trigonometric functions exhibit a beautiful symmetry and predictability, showing how calculus elegantly handles seemingly complex calculations with ease.

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Most popular questions from this chapter

Suppose that a piston is moving straight up and down and that its position at time \(t\) s is $$s=A \cos (2 \pi b t)$$ with \(A\) and \(b\) positive. The value of \(A\) is the amplitude of the motion, and \(b\) is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why some machinery breaks when you run it too fast.)

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I\). Perform the following steps: a. Plot the function \(f\) over \(I\) b. Find the linearization \(L\) of the function at the point \(a\) c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta>0\) as you can, satisfying \(|x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\epsilon\) for \(\epsilon=0.5,0.1,\) and \(0.01 .\) Then check graphically to see if your \(\delta\) -estimate holds true. $$f(x)=x^{3}+x^{2}-2 x, \quad[-1,2], \quad a=1$$

What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the functions. Find \(d y / d x\) if \(y=x\) by using the Chain Rule with \(y\) as a composite of a. \(y=(u / 5)+7\) and \(u=5 x-35\) b. \(y=1+(1 / u)\) and \(u=1 /(x-1)\)

Find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=\frac{2 u}{u^{2}+1}, \quad u=g(x)=10 x^{2}+x+1, \quad x=0$$

Find \(d y / d t\) $$y=(1+\cot (t / 2))^{-2}$$
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