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Chapter 16: Problem 72

How are the derivative and antiderivative of \(\sin x\) related?

Short Answer

Expert verified
The derivative of \(\sin x\) is \(\cos x\) and one of its antiderivatives is \(-\cos x\). Both functions are related to the cosine function but with opposite signs, indicating the derivative provides the tangent of the angle x, and the antiderivative gives the area under the curve of the sine function, up to a constant term.

Step by step solution

01

Find the derivative of \(\sin x\)

To find the derivative of the given function, we use the following rule: \(\frac{d}{dx}\sin x = \cos x\). So, the derivative of \(\sin x\) is \(\cos x\).
02

Find an antiderivative of \(\sin x\)

To find an antiderivative of \(\sin x\), we need to find a function whose derivative is equal to \(\sin x\). Using the derivative rules, we know that the derivative of \(-\cos x\) is \(\sin x\). Therefore, one of the antiderivatives of \(\sin x\) is \(-\cos x\).
03

Analyze the relationship

The derivative of \(\sin x\) is \(\cos x\) and one of its antiderivatives is \(-\cos x\). The relationship is that both functions are related to the cosine function but with opposite signs. The derivative provides the tangent of the angle x while the antiderivative gives the area under the curve of the sine function, up to a constant term.

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Key Concepts

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

Derivative
Understanding the derivative of a function is like grasping its rate of change. In calculus, when we say we are finding the derivative of \( \sin x\ \), we are looking at how the sine function changes at any point along its curve. It's a fundamental part of calculus and helps us see the immediate rate of change or slope of a function at any given point. Here's the important part for \( \sin x\ \): its derivative is \( \cos x\ \). This means at every point \( x \), the slope of the curve of \( \sin x\ \) is given by the cosine of that point.
  • The derivative helps in knowing how fast the function is changing.
  • It's calculated using specific rules 鈥� for \( \sin x\ \), that rule gives us \( \cos x\ \).
Knowing how to find derivatives helps in many real-world problems where change is a factor.
Antiderivative
Now, switching gears to antiderivatives, these are essentially the reverse of derivatives. When finding an antiderivative, we are looking for a function whose derivative would give us the original function we started with. For \( \sin x\ \), one of its antiderivatives is \( - \cos x\ \).
  • Antiderivatives help us find the area under a curve and are central to integral calculus.
  • For each function, there can be multiple antiderivatives, differing by a constant.
This concept is crucial for calculating total accumulated change over an interval, like total distance traveled.
Trigonometric Functions
Trigonometric functions are like the building blocks of many mathematical calculations, especially in calculus. They describe relationships in triangles and circles, making them very useful in oscillatory functions, angles, and waves.
  • \( \sin x\ \) represents the sine function, which plots a wave-like curve, important in modeling cyclical events.
  • Trigonometric derivatives and antiderivatives tell us about instantaneous changes and accumulated changes in these wave patterns.
Understanding these functions is crucial not just for calculus, but for physics, engineering, and many applied sciences.
Relationship Between Functions
The interplay between derivatives and antiderivatives underscores an important calculus principle: functions are deeply interconnected through these operations. For \( \sin x\ \), the derivative is \( \cos x\ \), and \( - \cos x\ \) serves as an antiderivative.
  • This relationship shows how the change in \( \sin x\ \) (its derivative) and the accumulation of \( \sin x\ \) (its antiderivative) are two sides of the same coin.
  • Recognizing this helps with solving complex calculus problems, as you can switch between the instantaneous and cumulative views.
These relationships highlight the elegant symmetry within calculus, linking different problem-solving techniques together.

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Most popular questions from this chapter

Evaluate the integrals. \(\int_{0.5}^{(\pi+1) / 2} \sin (2 x-1) d x\)

Calculate the derivatives. \( \frac{d}{d x}\left(\frac{\tan x}{2+e^{x}}\right)^{2}\)

Rectifier A rectifier is a circuit that converts alternating current to direct current. A full wave rectifier does so by effectively converting the voltage to its absolute value as a function of time. A 110 -volt 50 cycles per second \(\mathrm{AC}\) current would be converted to a voltage as shown: Compute and graph \(\frac{d V}{d t}\), and explain the sudden jumps you see in the graph of the derivative.

Evaluate the integrals. \(\int(4 x) \tan \left(x^{2}\right) d x\)

Use the shortcut formulas on page 1171 to calculate the integrals mentally. \(\int \cos (-x+1) d x\)
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