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3.6E: Exercises for Section 3.5

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    In exercises 1 - 10, find \(\dfrac{dy}{dx}\) for the given functions.

    1) \(y=x^2−\sec x+1\)

    \(\dfrac{dy}{dx}=2x−\sec x\tan x\)

    2) \(y=3\csc x+\dfrac{5}{x}\)

    3) \(y=x^2\cot x\)

    \(\dfrac{dy}{dx}=2x\cot x−x^2\csc^2 x\)

    4) \(y=x−x^3\sin x\)

    5) \(y=\dfrac{\sec x}{x}\)

    \(\dfrac{dy}{dx}=\dfrac{x\sec x\tan x−\sec x}{x^2}\)

    6) \(y=\sin x\tan x\)

    7) \(y=(x+\cos x)(1−\sin x)\)

    \(\dfrac{dy}{dx}=(1−\sin x)(1−\sin x)−\cos x(x+\cos x)\)

    8) \(y=\dfrac{\tan x}{1−\sec x}\)

    9) \(y=\dfrac{1−\cot x}{1+\cot x}\)

    \(\dfrac{dy}{dx}=\dfrac{2\csc^2 x}{(1+\cot x)^2}\)

    10) \(y=(\cos x)(1+\csc x)\)

    In exercises 11 - 16, find the equation of the tangent line to each of the given functions at the indicated values of \(x\). Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

    11) [T] \(f(x)=−\sin x,\quad x=0\)



    The graph shows negative sin(x) and the straight line T(x) with slope −1 and y intercept 0.

    12) [T] \(f(x)=\csc x,\quad x=\frac{π}{2}\)

    13) [T] \(f(x)=1+\cos x,\quad x=\frac{3π}{2}\)



    The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 – 3π)/2.

    14) [T] \(f(x)=\sec x,\quad x=\frac{π}{4}\)

    15) [T] \(f(x)=x^2−\tan x, \quad x=0\)



    The graph shows the function as starting at (−1, 3), decreasing to the origin, continuing to slowly decrease to about (1, −0.5), at which point it decreases very quickly.

    16) [T] \(f(x)=5\cot x, \quad x=\frac{π}{4}\)

    In exercises 17 - 22, find \(\dfrac{d^2y}{dx^2}\) for the given functions.

    17) \(y=x\sin x−\cos x\)

    \(\dfrac{d^2y}{dx^2} = 3\cos x−x\sin x\)

    18) \(y=\sin x\cos x\)

    19) \(y=x−\frac{1}{2}\sin x\)

    \(\dfrac{d^2y}{dx^2} = \frac{1}{2}\sin x\)

    20) \(y=\dfrac{1}{x}+\tan x\)

    21) \(y=2\csc x\)

    \(\dfrac{d^2y}{dx^2} = 2\csc(x)\left(\csc^2(x)+\cot^2(x)\right) \)

    22) \(y=\sec^2 x\)

    23) Find all \(x\) values on the graph of \(f(x)=−3\sin x\cos x\) where the tangent line is horizontal.

    \(x = \dfrac{(2n+1)π}{4}\),where \(n\) is an integer

    24) Find all \(x\) values on the graph of \(f(x)=x−2\cos x\) for \(0<x<2π\) where the tangent line has slope 2.

    25) Let \(f(x)=\cot x.\) Determine the points on the graph of \(f\) for \(0<x<2π\) where the tangent line(s) is (are) parallel to the line \(y=−2x\).

    \(\left(\frac{π}{4},1\right),\quad \left(\frac{3π}{4},−1\right),\quad\left(\frac{5π}{4},1\right),\quad \left(\frac{7π}{4},−1\right)\)

    26) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function \(s(t)=−6\cos t\) where s is measured in inches and \(t\) is measured in seconds. Find the rate at which the spring is oscillating at \(t=5\) s.

    27) Let the position of a swinging pendulum in simple harmonic motion be given by \(s(t)=a\cos t+b\sin t\). Find the constants \(a\) and \(b\) such that when the velocity is 3 cm/s, \(s=0\) and \(t=0\).

    \(a=0,\quad b=3\)

    28) After a diver jumps off a diving board, the edge of the board oscillates with position given by \(s(t)=−5\cos t\) cm at \(t\) seconds after the jump.

    a. Sketch one period of the position function for \(t≥0\).

    b. Find the velocity function.

    c. Sketch one period of the velocity function for \(t≥0\).

    d. Determine the times when the velocity is \(0\) over one period.

    e. Find the acceleration function.

    f. Sketch one period of the acceleration function for \(t≥0\).

    29) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by \(y=10+5\sin x\) where \(y\) is the number of hamburgers sold and \(x\) represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find \(y'\) and determine the intervals where the number of burgers being sold is increasing.

    \(y′=5\cos(x)\), increasing on \(\left(0,\frac{π}{2}\right),\;\left(\frac{3π}{2},\frac{5π}{2}\right)\), and \(\left(\frac{7π}{2},12\right)\)

    30) [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by \(y(t)=0.5+0.3\cos t\), where \(t\) is months since January. Find \(y′\)and use a calculator to determine the intervals where the amount of rain falling is decreasing.

    For exercises 31 - 33, use the quotient rule to derive the given equations.

    31) \(\dfrac{d}{dx}(\cot x)=−\csc^2x\)

    32) \(\dfrac{d}{dx}(\sec x)=\sec x\tan x\)

    33) \(\dfrac{d}{dx}(\csc x)=−\csc x\cot x\)

    34) Use the definition of derivative and the identity \(\cos(x+h)=\cos x\cos h−\sin x\sin h\) to prove that \(\dfrac{d}{dx}(\cos x)=−\sin x\).

    For exercises 35 - 39, find the requested higher-order derivative for the given functions.

    35) \(\dfrac{d^3y}{dx^3}\) of \(y=3\cos x\)

    \(\dfrac{d^3y}{dx^3} = 3\sin x\)

    36) \(\dfrac{d^2y}{dx^2}\) of \(y=3\sin x+x^2\cos x\)

    37) \(\dfrac{d^4y}{dx^4}\) of \(y=5\cos x\)

    \(\dfrac{d^4y}{dx^4} = 5\cos x\)

    38) \(\dfrac{d^2y}{dx^2}\) of \(y=\sec x+\cot x\)

    39) \(\dfrac{d^3y}{dx^3}\) of \(y=x^{10}−\sec x\)

    \(\dfrac{d^3y}{dx^3} = 720x^7−5\tan(x)\sec^3(x)−\tan^3(x)\sec(x)\)

    3.6E: Exercises for Section 3.5 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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