Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 27

Differentiation of Transcendental Functions - all with Video Answers

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Section 1

Derivatives of the sine and Cosine Functions

Problem 1

Make the given changes in the indicated examples of this section and then find the derivatives.
In Example $2,$ in the given function, change $\theta$ to $2 \theta^{2}$

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Problem 2

Make the given changes in the indicated examples of this section and then find the derivatives.
In Example $4,$ in the given function, change $2 x$ to $x^{2}$.

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Problem 3

Find the derivatives of the given functions.
$$y=\sin (3 x+2)$$

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Problem 4

Find the derivatives of the given functions.
$$y=3 \sin 4 x$$

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Problem 5

Find the derivatives of the given functions.
$$y=2 \sin \left(2 x^{3}-1\right)$$

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Problem 6

Find the derivatives of the given functions.
$$s=5 \sin (7-3 t)$$

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Problem 7

Find the derivatives of the given functions.
$$y=9 \cos \frac{4}{3} x$$

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Problem 8

Find the derivatives of the given functions.
$$y=\cos (1-2 x)$$

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Problem 9

Find the derivatives of the given functions.
$$y=3 x+2 \cos (3 x-\pi)$$

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Problem 10

Find the derivatives of the given functions.
$$y=x^{2}-\cos (1-3 x)$$

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Problem 11

Find the derivatives of the given functions.
$$r=\sin ^{2} 5 \pi \theta$$

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Problem 12

Find the derivatives of the given functions.
$$y=3 \sin ^{3}\left(2 x^{4}+1\right)$$

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Problem 13

Find the derivatives of the given functions.
$$y=3 \cos ^{3}(5 x+2)$$

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Problem 14

Find the derivatives of the given functions.
$$y=4 \cos ^{2} \sqrt{x}$$

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Problem 15

Find the derivatives of the given functions.
$$y=4 x \sin 3 x$$

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Problem 16

Find the derivatives of the given functions.
$$v=6 t^{2} \sin 3 \pi t$$

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Problem 17

Find the derivatives of the given functions.
$$y=3 x^{3} \cos 5 x$$

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Problem 18

Find the derivatives of the given functions.
$$y=0.5 \theta \cos (2 \theta+\pi / 4)$$

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Problem 19

Find the derivatives of the given functions.
$$u=3 \sin v^{2} \cos 5 v$$

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Problem 20

Find the derivatives of the given functions.
$$y=\sin 3 x \cos 4 x$$

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Problem 21

Find the derivatives of the given functions.
$$y=\sqrt{1+\sin 4 x}$$

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Problem 22

Find the derivatives of the given functions.
$$y=\left(3 x-\cos ^{2} x\right)^{4}$$

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Problem 23

Find the derivatives of the given functions.
$$r=\frac{\sin (3 t-\pi / 3)}{2 t}$$

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Problem 24

Find the derivatives of the given functions.
$$T=\frac{4 z+3}{\sin \pi z}$$

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Problem 25

Find the derivatives of the given functions.
$$y=\frac{2 \cos x^{2}}{3 x-1}$$

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Problem 26

Find the derivatives of the given functions.
$$y=\frac{\cos ^{2} 3 x}{1+2 \sin ^{2} 2 x}$$

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Problem 27

Find the derivatives of the given functions.
$$y=2 \sin ^{2} 3 x \cos 2 x$$

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Problem 28

Find the derivatives of the given functions.
$$y=\cos ^{3} 4 x \sin ^{2} 2 x$$

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Problem 29

Find the derivatives of the given functions.
$$s=\sin (3 \sin 2 t)$$

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Problem 30

Find the derivatives of the given functions.
$$z=0.2 \cos (4 \sin 3 \phi)$$

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Problem 31

Find the derivatives of the given functions.
$$y=\sin ^{3} x-\cos 2 x$$

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Problem 32

Find the derivatives of the given functions.
$$y=x \sin x+\cos x$$

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Problem 33

Find the derivatives of the given functions.
$$p=\frac{1}{\sin s}+\frac{1}{\cos s}$$

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Problem 34

Find the derivatives of the given functions.
$$y=2 x \sin x+2 \cos x-x^{2} \cos x$$

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Problem 35

Find the derivatives of the given functions.
$$\cos y=x^{2}+y$$

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Problem 36

Find the derivatives of the given functions.
$$y \sin x=2 y$$

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Problem 37

Solve the given problems.
Using a calculator: (a) display the graph of $y=(\sin x) / x$ to verify that $(\sin \theta) / \theta \rightarrow 1$ as $\theta \rightarrow 0,$ and (b) verify the values for $(\sin \theta) / \theta$ in the table on page 806

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Problem 38

Solve the given problems.
Evaluate $\left.\lim _{\theta \rightarrow 0}(\tan \theta) / \theta . \text { [Use the fact that } \lim _{\theta \rightarrow 0}(\sin \theta) / \theta=1 .\right]$

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Problem 39

Solve the given problems.
On a calculator, find the values of (a) $\cos 1.0000$ and
(b) $(\sin 1.0001-\sin 1.0000) / 0.0001 .$ Compare the values and
give the meaning of each in relation to the derivative of the sine function where $x=1$

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Problem 40

Solve the given problems.
On a calculator. find the values of (a) $-\sin 1.0000$ and (b) $(\cos 1.0001-\cos 1.0000) / 0.0001 .$ Compare the values
and give the meaning of each in relation to the derivative of the cosine function where $x=1$

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Problem 41

Solve the given problems.
On the graph of $y=\sin x$ in Fig. $27.3,$ draw tangent lines at the indicated points and estimate the slopes of these tangent lines (the slope at the $x$ -intercepts is ±1 ). Then plot the values of these slopes for the same values of $x$ and join the points with a smooth curve. Compare the resulting curve with $y=\cos x$. (Note the meaning of the derivative as the slope of a tangent line.)

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Problem 42

Solve the given problems.
Display the graphs of $y_{1}=\cos x$ and $y_{n}=\frac{\sin (x+h)-\sin x}{h}$ on
the same screen of a calculator for $0 \leq x \leq 2 \pi$. For $n=2$, let $h=0.5 ;$ for $n=3,$ let $h=0.1 .$ (You might try some even smaller values of $h$ ) What do these curves show?

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Problem 43

Solve the given problems.
Find the derivative $\frac{d y}{d x}$ of the implicit function $\sin (x y)+\cos 2 y=x^{2}$

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Problem 44

Solve the given problems.
Find the derivative $\frac{d y}{d x}$ of the implicit function $x \cos 2 y+\sin x \cos y=1$

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Problem 45

Solve the given problems.
Show that $\frac{d^{4} \sin x}{d x^{4}}=\sin x$

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Problem 46

Solve the given problems.
Show that $y=A \sin k x+B \cos k x$ satisfies the equation $y^{n}+k^{2} y=0$

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Problem 47

Solve the given problems.
Find the derivative of each member of the identity $\cos 2 x=2 \cos ^{2} x-1$ and thereby obtain another trigonometric identity.

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Problem 48

Solve the given problems.
Find values of $x$ for which the following curves have horizontal tangents: (a) $y=x+\sin x$ (b) $y=4 x+\cos \pi x$

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Problem 49

Solve the given problems.
Use differentials to estimate the value of $\sin 31^{\circ}$

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Problem 50

Solve the given problems.
Find the linearization $L(x)$ of the function $f(x)=\sin (\cos x)$ for $a=\pi / 2$

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Problem 51

Solve the given problems.
Find the slope of a line tangent to the curve of $y=\frac{2 \sin 3 x}{x},$ where $x=0.15 .$ Verify the result by using the derivative-evaluating feature of a calculator.

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Problem 52

Solve the given problems.
An object is oscillating vertically on the end of a spring such that its displacement $d$ (in $\mathrm{cm}$ ) is $d=2.5 \mathrm{cos} 16 t,$ where $t$ is the time (in s). What is the acceleration of the object after $1.5 \mathrm{s} ?$

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Problem 53

Solve the given problems.
The displacement $d$ (in $\mathrm{mm}$ ) of a piano wire as a function of the time $t$ (in s) is $d=3.0 \sin 188 t \cos 188 t .$ How fast is the displacement changing when $t=2.0 \mathrm{ms} ?$

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Problem 54

Solve the given problems.
A water slide at an amusement park follows the curve $(y \text { in } m$ ) $y=2.0+2.0 \cos (0.53 x+0.40)$ for $0 \leq x \leq 5.0 \mathrm{m} .$ Find the
angle with the horizontal of the slide for $x=2.5 \mathrm{m}$

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Problem 55

Solve the given problems.
The blade of a saber saw moves vertically up and down, and its displacement $y$ (in $\mathrm{cm}$ ) is given by $y=1.85$ sin $36 \pi t$ where $t$ is the time (in s). Find the velocity of the blade for $t=0.0250 \mathrm{s}$

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Problem 56

Solve the given problems.
The current $i$ (in $A$ ) in an amplifier circuit as a function of the time $t$ (in s) is given by $i=0.10 \cos (120 \pi t+\pi / 6) .$ The voltage caused by the changing current is given by $V_{L}=L(d i / d t),$ where $L$ is the inductance (in $\mathrm{H}$ ). Find the expression for the voltage across a $2.0-\mathrm{mH}$ inductor in the circuit.

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Problem 57

Solve the given problems.
In testing a heat-seeking rocket, it is always moving directly toward a remote-controlled aircraft. At a certain instant, the distance $r(\text { in } \mathrm{km})$ from the rocket to the aircraft is $r=\frac{100}{1-\cos \theta},$ where
$\theta$ is the angle between their directions of flight. Find $d r / d \theta$ for $\theta=120^{\circ} .$ See Fig. 27.4

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Problem 58

Solve the given problems.
The number $N$ of reflections of a light ray passing through an optic fiber of length $L$ and diameter $d$ is $N=\frac{L \sin \theta}{d \sqrt{n^{2}-\sin ^{2} \theta}} .$ Here, $n$ is the index of refraction of the fiber, and $\theta$ is the angle between the light ray and the fiber's axis. Find $d N / d \theta$

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Problem 59

Solve the given problems.
In the study of the transmission of light, the equation $T=\frac{A}{1+B \sin ^{2}(\theta / 2)}$ arises. Find $d T / d \theta$

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Problem 60

Solve the given problems.
A conical paper cup has a slant height of $10.0 \mathrm{cm}$. Express the volume $V$ of the cup in terms of $\theta / 2,$ where $\theta$ is the vertex angle at the bottom of the cup. Then find $d V / d \theta$

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