1
Differentiate.
\(f(x) = 3 \sin x - 2 \cos x\)
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2
Differentiate.
\(f(x) = \tan x - 4 \sin x\)
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3
Differentiate.
\(y = x^2 + \cot x\)
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4
Differentiate.
\(y = 2 \sec x - \csc x\)
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5
Differentiate.
\(h(\theta) = \theta^2 \sin \theta\)
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6
Differentiate.
\(g(x) = 3x + x^2 \cos x\)
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7
Differentiate.
\(y = \sec \theta \tan \theta\)
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8
Differentiate.
\(y = \sin \theta \cos \theta\)
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9
Differentiate.
\(f(\theta) = (\theta - \cos \theta) \sin \theta\)
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10
Differentiate.
\(f(x) = x \cos x + 2 \tan x\)
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11
Differentiate.
\(H(t) = \cos^2 t\)
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12
Differentiate.
\(y = u(a \cos u + b \cot u)\)
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13
Differentiate.
\(f(\theta) = \dfrac{\sin \theta}{1 + \cos \theta}\)
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14
Differentiate.
\(y = \dfrac{\cos x}{1 - \sin x}\)
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15
Differentiate.
\(y = \dfrac{x}{2 - \tan x}\)
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16
Differentiate.
\(f(t) = \dfrac{\cot t}{t^2}\)
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17
Differentiate.
\(f(w) = \dfrac{1 + \sec w}{1 - \sec w}\)
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18
Differentiate.
\(y = \dfrac{\sin t}{1 + \tan t}\)
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19
Differentiate.
\(y = \dfrac{t \sin t}{1 + t}\)
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20
Differentiate.
\(g(z) = \dfrac{z}{\sec z + \tan z}\)
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21
Differentiate.
\(f(\theta) = \theta \cos \theta \sin \theta\)
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22
Differentiate.
\(y = x^2 \sin x \tan x\)
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23
Show that \(\dfrac{d}{d x}(\csc x) = -\csc x \cot x\).
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24
Show that \(\dfrac{d}{d x}(\sec x) = \sec x \tan x\).
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25
Show that \(\dfrac{d}{d x}(\cot x) = -\csc^2 x\).
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26
Prove, using the definition of derivative, that if \(f(x) = \cos x\), then \(f'(x) = -\sin x\).
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27
Find an equation of the tangent line to the curve at the given point.
\(y = \sin x + \cos x\), quad \((0, 1)\)
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28
Find an equation of the tangent line to the curve at the given point.
\(y = x + \sin x\), quad \((\pi, \pi)\)
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29
Find an equation of the tangent line to the curve at the given point.
\(y = x + \tan x\), quad \((\pi, \pi)\)
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30
Find an equation of the tangent line to the curve at the given point.
\(y = \dfrac{1 + \sin x}{\cos x}\), quad \((\pi, -1)\)
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31
(a) Find an equation of the tangent line to the curve \(y = 2x \sin x\) at the point \(\left(\dfrac{\pi}{2}, \pi\right)\).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
Enter your answer directly below each part above.
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32
(a) Find an equation of the tangent line to the curve \(y = 3x + 6 \cos x\) at the point \(\left(\dfrac{\pi}{3}, \pi + 3\right)\).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
Enter your answer directly below each part above.
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33
(a) If \(f(x) = \sec x - x\), find \(f'(x)\).
(b) Check your answer to part (a) by graphing both \(f\) and \(f'\) for \(|x| < \dfrac{\pi}{2}\).
Enter your answer directly below each part above.
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34
(a) If \(f(x) = \sqrt{x} \sin x\), find \(f'(x)\).
(b) Check your answer to part (a) by graphing both \(f\) and \(f'\) for \(0 \leq x \leq 2 \pi\).
Enter your answer directly below each part above.
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35
If \(g(\theta) = \dfrac{\sin \theta}{\theta}\), find \(g'(\theta)\) and \(g''(\theta)\).
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36
If \(f(t) = \sec t\), find \(f''\left(\dfrac{\pi}{4}\right)\).
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37
(a) Use the Quotient Rule to differentiate the function \(f(x) = \dfrac{\tan x - 1}{\sec x}\).
(b) Simplify the expression for \(f(x)\) by writing it in terms of \(\sin x\) and \(\cos x\), and then find \(f'(x)\).
(c) Show that your answers to parts (a) and (b) are equivalent.
Enter your answer directly below each part above.
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38
Suppose \(f\left(\dfrac{\pi}{3}\right) = 4\) and \(f'\left(\dfrac{\pi}{3}\right) = -2\), and let \(g(x) = f(x) \sin x\) and \(h(x) = \dfrac{\cos x}{f(x)}\). Find:
(a) \(g'\left(\dfrac{\pi}{3}\right)\)
(b) \(h'\left(\dfrac{\pi}{3}\right)\)
Enter your answer directly below each part above.
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39
For what values of \(x\) does the graph of \(f(x) = x + 2 \sin x\) have a horizontal tangent?
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40
Find the points on the curve \(y = \dfrac{\cos x}{2 + \sin x}\) at which the tangent is horizontal.
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41
An object with mass \(m\) is attached to the end of a spring that is stretched and then released. Its displacement from its equilibrium position at time \(t\) is \(x(t) = 8 \sin t\).
(a) Find the velocity and acceleration at time \(t\).
(b) Find the position, velocity, and acceleration at time \(t = \dfrac{2 \pi}{3}\). In which direction is the object moving at that time?
(c) When does the object first pass through its equilibrium position?
(d) How far from its equilibrium position does the object travel?
(e) When is the speed greatest?
Enter your answer directly below each part above.
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42
An elastic band is hung from a hook and a mass is attached to the lower end. When the mass is pulled downward and released, it vibrates vertically. The equation of motion is \(s = 2 \cos t + 3 \sin t\), \(t \geq 0\), where \(s\) is measured in centimeters and \(t\) in seconds.
(a) Find the velocity and acceleration at time \(t\).
(b) Graph the velocity and acceleration functions.
(c) When does the mass first pass through the equilibrium position going downward?
(d) How far from its equilibrium position does the mass travel?
(e) When is the speed greatest?
Enter your answer directly below each part above.
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43
A ladder \(10\) ft long leans against a vertical wall. Let \(\theta\) be the angle between the top of the ladder and the wall, and let \(x\) be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does \(x\) change with respect to \(\theta\) when \(\theta = \dfrac{\pi}{3}\)?
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44
An object with weight \(W\) is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle \(\theta\) with the plane, then the magnitude of the force is
\( F = \dfrac{\mu W}{\mu \sin \theta + \cos \theta} \)
where \(\mu\) is the coefficient of friction.
(a) Find the rate of change of \(F\) with respect to \(\theta\).
(b) When is this rate of change equal to \(0\)?
(c) If \(W = 50\) lb and \(\mu = 0.6\), draw the graph of \(F\) as a function of \(\theta\) and use it to locate the value of \(\theta\) for which \(\dfrac{d F}{d \theta} = 0\). Is the value consistent with your answer to part (b)?
Enter your answer directly below each part above.
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45
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin 5x}{3x} \)
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46
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin x}{\sin \pi x} \)
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47
Find the limit.
\( \operatorname*{lim}\limits_{t \rightarrow 0} \dfrac{\sin 3t}{\sin t} \)
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48
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin^2 3x}{x} \)
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49
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin x - \sin x \cos x}{x^2} \)
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50
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{1 - \sec x}{2x} \)
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51
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\tan 2x}{x} \)
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52
Find the limit.
\( \operatorname*{lim}\limits_{\theta \rightarrow 0} \dfrac{\sin \theta}{\tan 7 \theta} \)
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53
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin 3x}{5x^3 - 4x} \)
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54
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin 3x \sin 5x}{x^2} \)
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55
Find the limit.
\( \operatorname*{lim}\limits_{\theta \rightarrow 0} \dfrac{\sin \theta}{\theta + \tan \theta} \)
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56
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \csc x \sin(\sin x) \)
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57
Find the limit.
\( \operatorname*{lim}\limits_{\theta \rightarrow 0} \dfrac{\cos \theta - 1}{2 \theta^2} \)
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58
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{\sin(x^2)}{x} \)
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59
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{4}} \dfrac{1 - \tan x}{\sin x - \cos x} \)
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60
Find the limit.
\( \operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{\sin(x - 1)}{x^2 + x - 2} \)
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61
Find the given derivative by finding the first few derivatives and observing the pattern that occurs.
\( \dfrac{d^{99}}{d x^{99}}(\sin x) \)
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62
Find the given derivative by finding the first few derivatives and observing the pattern that occurs.
\( \dfrac{d^{35}}{d x^{35}}(x \sin x) \)
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63
Find constants \(A\) and \(B\) such that the function \(y = A \sin x + B \cos x\) satisfies the differential equation \(y'' + y' - 2y = \sin x\).
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64
Evaluate \(\operatorname*{lim}\limits_{x \rightarrow 0} x \sin\left(\dfrac{1}{x}\right)\) and illustrate by graphing the function \(y = x \sin\left(\dfrac{1}{x}\right)\).
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65
Differentiate each trigonometric identity to obtain a new (or familiar) identity.
(a) \(\tan x = \dfrac{\sin x}{\cos x}\)
(b) \(\sec x = \dfrac{1}{\cos x}\)
(c) \(\sin x + \cos x = \dfrac{1 + \cot x}{\csc x}\)
Enter your answer directly below each part above.
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66
The figure shows a semicircle with diameter \(P Q\) sitting on an isosceles triangle \(P Q R\) to form a region shaped like an ice-cream cone. If \(A(\theta)\) is the area of the semicircle and \(B(\theta)\) is the area of the triangle, find
\( \operatorname*{lim}\limits_{\theta \rightarrow 0^+} \dfrac{A(\theta)}{B(\theta)} \)
The sides of the triangle have length \(10\) cm.
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67
The figure shows a circular arc of length \(s\) and a chord of length \(d\), both subtended by a central angle \(\theta\). Find
\( \operatorname*{lim}\limits_{\theta \rightarrow 0^+} \dfrac{s}{d} \)
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68
Let \(f(x) = \dfrac{x}{\sqrt{1 - \cos 2x}}\).
(a) Graph \(f\). What type of discontinuity does it appear to have at \(0\)?
(b) Calculate the left and right limits of \(f\) at \(0\). Do these values confirm your answer to part (a)?
Enter your answer directly below each part above.
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Answered: 0 / 68