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Chapter 3: Problem 80

Prove that $$\frac{d^{2 n}}{d x^{2 n}}(\sin x)=(-1)^{n} \sin x \text { and } \frac{d^{2 n}}{d x^{2 n}}(\cos x)=(-1)^{n} \cos x.$$

Short Answer

Expert verified
Question: Prove the expressions $$\frac{d^{2n}}{dx^{2n}}(\sin x) = (-1)^n \sin x$$ and $$\frac{d^{2n}}{dx^{2n}}(\cos x) = (-1)^n \cos x$$. Answer: By using the derivative formulas for sine and cosine functions and repeatedly applying the differentiation rule, we identified a pattern in the derivatives and successfully proved that $$\frac{d^{2n}}{dx^{2n}}(\sin x) = (-1)^n \sin x$$ and $$\frac{d^{2n}}{dx^{2n}}(\cos x) = (-1)^n \cos x$$.

Step by step solution

01

Identify the formulas for sine and cosine derivatives

To solve this problem, we need to remember the derivative formulas for sine and cosine functions: $$ \frac{d}{dx}(\sin x) = \cos x,\quad \frac{d}{dx}(\cos x) = -\sin x $$
02

Differentiation of sine function

We want to find $$\frac{d^{2n}}{dx^{2n}}(\sin x)$$, which means taking the derivative of the sin(x) function 2n times. We will do this by repeatedly differentiating the sine function: 1st derivative: $$\frac{d}{dx}(\sin x) = \cos x$$ 2nd derivative: $$\frac{d^2}{dx^2}(\sin x) = \frac{d}{dx}(\cos x)=-\sin x =(-1)^1 \sin x$$ Now, let's find a pattern in the derivatives: 3rd derivative: $$\frac{d^3}{dx^3}(\sin x) = \frac{d^2}{dx^2}(\cos x)=(-1)^1 \cos x$$ 4th derivative: $$\frac{d^4}{dx^4}(\sin x) = \frac{d^3}{dx^3}(-\sin x)=(-1)^2 \sin x$$ From the pattern, we can identify that $$\frac{d^{2n}}{dx^{2n}}(\sin x)= (-1)^n \sin x$$
03

Differentiation of cosine function

Similarly, we want to find $$\frac{d^{2n}}{dx^{2n}}(\cos x)$$, which means taking the derivative of the cos(x) function 2n times. 1st derivative: $$\frac{d}{dx}(\cos x) = -\sin x$$ 2nd derivative: $$\frac{d^2}{dx^2}(\cos x) = \frac{d}{dx}(-\sin x)= -\cos x =(-1)^1 \cos x$$ Let's find a pattern in the derivatives like we did with the sine function: 3rd derivative: $$\frac{d^3}{dx^3}(\cos x) = \frac{d^2}{dx^2}(-\sin x)=(-1)^1 \sin x$$ 4th derivative: $$\frac{d^4}{dx^4}(\cos x) = \frac{d^3}{dx^3}(-\cos x)=(-1)^2 \cos x$$ From the pattern, we can identify that $$\frac{d^{2n}}{dx^{2n}}(\cos x)= (-1)^n \cos x$$
04

Conclusion

After examining the patterns of sequential derivatives for sine and cosine functions, we have successfully proven that $$\frac{d^{2n}}{dx^{2n}}(\sin x) = (-1)^n \sin x \text{ and } \frac{d^{2n}}{dx^{2n}}(\cos x) = (-1)^n \cos x$$

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Key Concepts

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

Derivatives of Sine and Cosine
Understanding the derivatives of sine and cosine functions is foundational to calculus, as these functions are prevalent in many areas of mathematics and physics. When you take the first derivative of the sine function, denoted as \( \frac{d}{dx}(\sin x) \), you get the cosine function, \( \cos x \). Conversely, the first derivative of the cosine function, \( \frac{d}{dx}(\cos x) \), yields negative sine, or \( -\sin x \).

Knowing these basic derivatives allows students to tackle more complex problems, including finding higher-order derivatives. When working with higher-order derivatives of these trigonometric functions, it's necessary to repeatedly differentiate, applying the fundamental derivatives over and over again.
Derivative Patterns
In calculus, identifying patterns in derivatives can greatly simplify the process of differentiation, especially when dealing with higher-order derivatives. As illustrated in the exercise, the process of differentiating \( \sin x \) and \( \cos x \) repeatedly reveals a cyclic pattern. For \( \sin x \), the derivatives alternate between \( \cos x \) and \( -\sin x \) and then back to \( -\cos x \) and \( \sin x \) for the second set of derivatives, and so on. The pattern repeats every four derivatives.

This tells us that the nth derivative will depend on whether n is even or odd and its position within the four-step cycle. When working with 2n derivatives, as in the exercise, we remain within the bounds of \( \sin x \) and \( \cos x \) functions, simply adjusting the sign based on the index. Utilizing these patterns can greatly reduce the computational effort and lends itself to mathematical proof, as seen in the textbook solution.
Mathematical Proof
The process of mathematical proof involves logical reasoning to establish the truth of a statement. In our textbook exercise, we are tasked with proving a statement about higher-order derivatives of sine and cosine functions. The proof builds on the observed pattern in derivatives and uses induction to generalize the outcome for any natural number, n.

By methodically demonstrating that the derivative patterns hold for the first few cases, we can imply the validity of the pattern for all subsequent natural numbers. In the case of the 2n-th derivative of both \( \sin x \) and \( \cos x \) functions, the pattern simplifies to multiplying by \( (-1)^n \) for any natural number n. This elegant result not only fulfills the need for mathematical rigor but also greatly simplifies many calculus problems involving trigonometric functions.

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Most popular questions from this chapter

Suppose a large company makes 25,000 gadgets per year in batches of \(x\) items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, it has been determined that the total cost \(C(x)\) of producing 25,000 gadgets in batches of \(x\) items at a time is given by $$C(x)=1,250,000+\frac{125,000,000}{x}+1.5 x.$$ a. Determine the marginal cost and average cost functions. Graph and interpret these functions. b. Determine the average cost and marginal cost when \(x=5000\). c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b).

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=c x^{2} ; x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants

Logistic growth Scientists often use the logistic growth function \(P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{d}}}\) to model population growth, where \(P_{0}\) is the initial population at time \(t=0, K\) is the carrying capacity, and \(r_{0}\) is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. World population (part 1 ) The population of the world reached 6 billion in \(1999(t=0)\). Assume Earth's carrying capacity is 15 billion and the base growth rate is \(r_{0}=0.025\) per year. a. Write a logistic growth function for the world's population (in billions) and graph your equation on the interval \(0 \leq t \leq 200\) using a graphing utility. b. What will the population be in the year 2020? When will it reach 12 billion?

a. Use derivatives to show that \(\tan ^{-1} \frac{2}{n^{2}}\) and \(\tan ^{-1}(n+1)-\tan ^{-1}(n-1)\) differ by a constant. b. Prove that \(\tan ^{-1} \frac{2}{n^{2}}=\tan ^{-1}(n+1)-\tan ^{-1}(n-1)\)

A cylindrical tank is full at time \(t=0\) when a valve in the bottom of the tank is opened. By Torricelli's Law, the volume of water in the tank after \(t\) hours is \(V=100(200-t)^{2}\), measured in cubic meters. a. Graph the volume function. What is the volume of water in the tank before the valve is opened? b. How long does it take the tank to empty? c. Find the rate at which water flows from the tank and plot the flow rate function. d. At what time is the magnitude of the flow rate a minimum? A maximum?
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