4.5 Chapter Review
Chapter
Chapter 4
Section
4.5
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Solutions 24 Videos

The graph of a sinusoidal function is as shown.

a) Identify the x-values in the interval 0 \leq x \leq 2\piwhere the slope is

i) zero

ii) a local maximum

iii) a local minimum

b) Sketch a graph of the instantaneous rate of change of this function with respect to x.

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Q1a

a) Sketch a graph of the function y= -3\sin x

b) Sketch a graph of the instantaneous rate of change of this function with respect to x.

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Q2

Find the derivative of each function with respect to x.

a) y=\cos x

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Q3a

Find the derivative of each function with respect to x.

b) f(x) = -2\sin x

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Q3b

Find the derivative of each function with respect to x.

c) y = \cos x - \sin x

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Q3c

Find the derivative of each function with respect to x.

d) f(x) = 3\sin x - \pi\cos x

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Q3d

Determine the slope of the function y = 4 \sin x at x= \dfrac{\pi}{3}

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Q4

a) Find the equation of the line that is tangent to the curve y = 2 \sin \theta + 4 \cos \theta at \theta = \dfrac{\pi}{4}.

b) Find the equation of the line that is tangent to to the curve y = 2 \cos \theta - \dfrac{1}{2}\sin \theta at \theta = \dfrac{3\pi}{2}

c) Graph each function and it's tangent lines from parts a) and b).

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Q5

Differentiate.

y = -\cos^2x

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Q6a

Differentiate.

y = \sin 2 \theta - 2\cos 2\theta

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Q6b

Differentiate.

f(\theta) = -\dfrac{\pi}{2} \sin (2\theta - \pi)

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Q6c

Differentiate.

f(x) = \sin (\sin x)

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Q6d

Differentiate.

f(x) = \cos (\cos x)

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Q6e

Differentiate.

f(\theta) = \cos 7\theta - \cos (5\theta)

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Q6f

Determine the derivative of each function.

y = 3x\sin x

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Q7a

Determine the derivative of each function.

f(t) = 2t^2\cos 2t

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Q7b

Determine the derivative of each function.

y = \pi t\sin (\pi t - 6)

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Q7c

Determine the derivative of each function.

y = \cos (\sin \theta) + \sin (cos \theta)

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Q7d

Determine the derivative of each function.

f(x) = \cos^2 (\sin x)

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Q7e

Determine the derivative of each function.

f(\theta) = \cos 7\theta - \cos ^2 5\theta

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Q7f

a) Find and equation of a line that is tangent to the curve f(x) = 2\cos 3x and whose slope is a maximum.

b) Is this the only possible solution? Explain. If there are other possible solutions, how many solutions are there?

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Q8

The voltage of the power supply in an AC-DC couple circuit is given by the function V(t) = 130 \sin 5t + 18, where t is time, in seconds, and V is the voltage in volts, at time t.

a) Find the maximum and minimum voltages and the times at which they occur.

b) Determine the period, T, in seconds, the frequency, f, in hertz, and the amplitude, A, in volts, for this signal.

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Q9

A block is positioned on a frictionless ramp as shown.

The gravitational force component acting on the block directed along the ramp as a function of the angle of inclination, \theta, is F= mg\sin \theta, where m is the mass of the block and g is the acceleration of gravity.

a) For what angle of inclination, 0 \leq \theta \leq \dfrac{\pi}{2}, will this force be

i) a maximum?

ii) a minimum?

b) Explain how you found your answers in part a). Explain why these answers make sense.

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Q10

Newton's second law of the motion was originally written as F= \dfrac{dp}{dt}, where

  • F is the force of acting on the object, in newtons

  • p is the momentum of the object, in kilogram metres per second, given by the equation p =mv (where m is the mass of the object, in kilograms, and v is the velocity of the object in metres per second).

  • t is time, in seconds.

a) Assuming an object's mass is constant, use this definition to show that Newton's second law can also be written as F = ma, where a is the acceleration of the object.

b) Suppose an object is oscillating such that its velocity, in metre per second, at time t is given by the function v(t) = 2\cos 3t, where t is time, in seconds. Find the times when the force acting on the object is zero.

c) What is the speed of the object at these times?

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Q11