4.6 Chapter Test
Chapter
Chapter 4
Section
4.6
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Solutions 22 Videos

Where do the minimum instantaneous rates of change occur for the function shown here?

A (2k +1) \pi, k \in \mathbb{Z}

B 2k\pi, k \in \mathbb{Z}

C \dfrac{\pi}{2} + k\pi, k \in \mathbb{Z}

D \dfrac{\pi}{2} + 2k\pi, k \in \mathbb{Z}

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Q1

If f(x) = \sin 3x + \cos 2x , then

A f'(x) = \cos 3x - \sin 2x

B f'(x) = 3\sin x + 2\cos x

C f'(x) = 3\cos 3x - 2\sin 2x

D f'(x) = 5\cos 5x

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Q2

If g(x) = 3\cos 2x, then

A g'(x) = 6\cos x

B g'(x) = -6\sin x

C g'(x) = -6\sin 2x

D g'(x) = -6\cos 2x

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Q3

If h(x) = x\sin x, then

A h'(x) = \cos x

B h'(x) = \sin x - \cos x

C h'(x) = \sin x + x - \cos x

D h'(x) = \sin x + x\cos x

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Q4

Which is the second derivative of the function y = -\cos x with respect to x?

A \dfrac{d^2y}{dx^2} = \sin x

B \dfrac{d^2y}{dx^2} = -\sin x

C \dfrac{d^2y}{dx^2} = \cos x

D \dfrac{d^2y}{dx^2} = -\cos x

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Q5

What is the slope of the curve y = 2\sin x at x= \dfrac {\pi}{3}

A - \sqrt{3}

B \sqrt {3}

C -1

D 1

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Q6

Which graph best shows the derivative of the function y = \sin x + \cos x when the window variables are x\in [-2\pi, 2\pi], Xscl = \dfrac{\pi}{2} , y = \in [-4, 4]?

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Q7

Which is the derivative of y = \cos \theta \sin \theta with respect to \theta?

A \dfrac{dy}{d\theta} = \cos ^2 \theta + \sin ^2 \theta

B \dfrac {dy}{d\theta} = \cos ^2 \theta - \sin ^2 \theta

C \dfrac {dy}{d\theta} = \sin ^2 \theta - \cos ^2 \theta

D \dfrac{dy}{d\theta} = -\sin ^2 \theta - \cos ^2

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Q8

Differentiate.

y = \cos x - \sin x

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Q9a

Differentiate.

y = 3\sin 2\theta

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Q9b

Differentiate.

f(x) = -\dfrac{\pi}{2} \cos ^2 x

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Q9c

Differentiate.

f(t) = 3t^2 \sin t

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Q9d

Differentiate with respect to \theta.

y = \sin (\theta + \dfrac{\pi}{4})

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Q10a

Differentiate with respect to \theta.

y = \cos (\theta - \dfrac {\pi}{4})

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Q10b

Differentiate with respect to \theta.

y = \sin ^4 \theta

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Q10c

Differentiate with respect to \theta.

y = \sin \theta ^4

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Q10d

Find the slope of the line that is tangent to the curve y = 2\sin x \cos x at x= \dfrac{\pi}{4}

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Q11

Find the equation of the line that is tangent to the curve y = 2\cos^3 x at x = \dfrac{\pi}{3}.

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Q12

The voltage signal from a standard European wall socket can be described by the equation V(t) = 325\sin 100\pi t , where t is time, in seconds, and V is the voltage at time t.

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

b) Determine

i) the period, T, in seconds

ii) the frequency, f, in hertz

iii) the amplitude, A, in volts

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Q13

Refer to the last question. Compare the standard wall socket voltage signals of Europe and North America. Recall that in North America, the voltage is described by the function V(t)= 170\sin 120\pi t

a) Repeat the last question using the function for the North American wall socket voltage.

b) Discuss the similarities and differences between these functions.

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Q14

Differentiate both sides of double angle identity \sin 2x = 2x\sin x \cos x to determine an identity for \cos 2x

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Q15

Let f(x) = \sin x \cos x .

a) Determine the first derivative, f'(x), and the second derivative, f"(x).

b) Determine the third, fourth, fifth, and sixth derivatives. Describe any patterns that you notice.

c) Make a prediction for the seventh and eighth derivatives. Calculate these and check your predictions.

d) Write an expression for each of the following derivatives of f(x):

  • i) f ^{2n}(x) , n \in \mathbb{N}
  • ii) f^{2n +1}(x), n \in \mathbb{N}

e) Use the results from the previous question to determine:

  • i) f^{12}(x)
  • ii) f^{15}(x)
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Q16