91Ó°ÊÓ

Slopes on sine curves a. Find equations for the tangents to the curves \(y=\sin 2 x\) and \(y=-\sin (x / 2)\) at the origin. Is there anything special about how the tangents are related? Give reasons for your answer. b. Can anything be said about the tangents to the curves \(y=\sin m x\) and \(y=-\sin (x / m)\) at the origin \((m\) a constant \(\neq 0) ?\) Give reasons for your answer. c. For a given \(m,\) what are the largest values the slopes of the curves \(y=\sin m x\) and \(y=-\sin (x / m)\) can ever have? Give reasons for your answer. d. The function \(y=\sin x\) completes one period on the interval \([0,2 \pi],\) the function \(y=\sin 2 x\) completes two periods, the function \(y=\sin (x / 2)\) completes half a period, and so on. Is there any relation between the number of periods \(y=\sin m x\) completes on \([0,2 \pi]\) and the slope of the curve \(y=\sin m x\) at the origin? Give reasons for your answer.

Step by step solution

01

Find the Derivative of y=sin(2x)

To find the equation of the tangent line at the origin, we need the slope of the tangent line. First, find the derivative of the function. For \(y = \sin 2x\), the derivative is \(\frac{dy}{dx} = 2 \cos 2x\). Since we are finding the slope at the origin \((x = 0)\), substitute \(x = 0\) into the derivative: \(\frac{dy}{dx} \bigg|_{x=0} = 2 \cos 0 = 2\). Thus, the slope is 2.

02

Find the Tangent Line Equation for y=sin(2x)

Using the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), with slope \(m = 2\) and the origin \((0,0)\), the equation is \(y = 2x\).

03

Find the Derivative of y=-sin(x/2)

Now find the derivative of \(y = -\sin(x/2)\). The derivative is \(\frac{dy}{dx} = -\frac{1}{2}\cos(x/2)\). Evaluate this at the origin \((x = 0)\): \(\frac{dy}{dx} \bigg|_{x=0} = -\frac{1}{2}\cos 0 = -\frac{1}{2}\). Thus, the slope is \(-\frac{1}{2}\).

04

Find the Tangent Line Equation for y=-sin(x/2)

Using the slope \(-\frac{1}{2}\) and the origin \((0,0)\), the equation of this tangent line is \(y = -\frac{1}{2}x\).

05

Analyze the Relationship Between the Tangents

The slopes at the origin are 2 for \(y=\sin 2x\) and \(-\frac{1}{2}\) for \(y=-\sin(x/2)\). These lines are perpendicular because the product of their slopes is \(-1\).

06

Generalize for y=sin(mx) and y=-sin(x/m)

Find the derivatives at the origin for \(y = \sin(mx)\) and \(y=-\sin(x/m)\). The slopes are \(m\) and \(-\frac{1}{m}\) respectively. Their relationship remains perpendicular since their product is \(-1\).

07

Determine Maximum Slopes

For \(y = \sin(mx)\), the maximum slope can be \(m\) when \(x=0\), and for \(y = -\sin(x/m)\), the maximum positive value is also \(m\) when \(m\) is positive and \(-m\) when \(m\) is negative, given by the magnitude of their derivatives \(m\).

08

Relate Periods to Slope at Origin

For \(y = \sin(mx)\), every period is \(\frac{2\pi}{m}\). More periods in \([0, 2\pi]\) implies a greater slope at the origin. This results from a more rapid oscillation due to the increased frequency.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions, sine and cosine, are used to describe periodic phenomena like sound waves, light waves, and tides.

They are fundamental in mathematics because:

• They help in understanding oscillating systems.
• They have a repetitive nature, known as periodicity.
• They travel in waves which correspond to cycles, circles, or patterns.

The sine function, denoted as \(y = \sin x\), and its variations, like \(y = \sin 2x\) or \(y = -\sin(x/2)\), are essential in this domain. They allow us to model how waves behave, showing peaks and troughs at various points.

Derivatives

Derivatives are a critical concept in calculus, representing the rate at which a function is changing at any point. In simpler terms, the derivative tells us the slope of a function at a particular point.

For trigonometric functions, derivatives give us vital information about the tangent lines, which are straight lines that touch curves at just one point. To find the tangent line’s slope at any point on the curve \(y = \sin 2x\), one must compute the derivative. For instance, the derivative \( \frac{dy}{dx} = 2\cos 2x \) shows how the slope varies with each \( x \) along the curve.

Moreover, understanding derivatives of more complicated functions like \(y = -\sin(x/2)\) helps in finding critical points and understanding curvature and inflection points of waves.

Periodic Functions

A function is periodic if it repeats its values in regular intervals or periods. Trigonometric functions like sine and cosine are the quintessential periodic functions, as their values repeat every specific interval, usually \(2\pi\) or \(\pi\) radians.

The function \(y = \sin x\) repeats every \(2\pi\), but adding a coefficient, such as \(m\) in \(y = \sin(mx)\), alters the function's period, making it \(\frac{2\pi}{m}\). This adjustment changes the frequency and how often the pattern repeats over a given range, affecting the overall appearance and behavior of the resulting wave.

Understanding periodic functions is crucial as they describe many natural phenomena, including musical tones, light waves, and the day-night cycle.

Frequency in Mathematics

Frequency in mathematics, particularly in the study of trigonometric functions, refers to the number of times a function repeats its pattern over a set period. Higher frequency means more cycles within the same range, leading to more oscillations.

In equations like \(y = \sin(mx)\), \(m\) is the frequency. It determines how fast the sine wave oscillates. Higher values of \(m\) mean the function completes more cycles within the interval \([0, 2\pi]\).

This concept is vital because it helps in understanding not just the function's geometry and shape, but also its application in describing real-world phenomena. Higher frequency can correspond to higher pitch sounds or more rapid fluctuations in an alternating current.