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 Differentiate the function f(x) =5e3x2\text { Differentiate the function } f ( x ) = 5 e ^ { 3 x ^ { 2 } }


A) 30xe6x30 x e ^ { 6 x }
B) 15xe3x215 x e ^ { 3 x ^ { 2 } }
C) 15e3x215 e ^ { 3 x ^ { 2 } }
D) 30xe3x230 x e ^ { 3 x ^ { 2 } }
E) 15e6x15 e ^ { 6 x }

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 Find ft(x)  if f(x) =log⁡5(x3−1x3−6) \text { Find } f ^ {t } ( x ) \text { if } f ( x ) = \log _ { 5 } \left( \frac { x ^ { 3 } - 1 } { x ^ { 3 } - 6 } \right)


A) ft(x) =x2−5(x3−1) (x2−6) f ^ { t } ( x ) = \frac { x ^ { 2 } - 5 } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 2 } - 6 \right) }
B) ft(x) =(x3−1) (x3−6) ln⁡(5) f ^ { t } ( x ) = \frac { \left( x ^ { 3 } - 1 \right) } { \left( x ^ { 3 } - 6 \right) \ln ( 5 ) }
C) ft(x) =−15x2(x3−1) (x3−6) f ^ { t } ( x ) = \frac { - 15 x ^ { 2 } } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 3 } - 6 \right) }
D) ft(x) =x3−6(x3−1) ln⁡(5) f ^ { t } ( x ) = \frac { x ^ { 3 } - 6 } { \left( x ^ { 3 } - 1 \right) \ln ( 5 ) }
E) ft(x) =−15x2(x3−1) (x3−6) ln⁡(5) f ^ { t } ( x ) = \frac { - 15 x ^ { 2 } } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 3 } - 6 \right) \ln ( 5 ) }

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 Find the derivative of the function f(x) =13arcsin⁡(x−1) . \text { Find the derivative of the function } f ( x ) = 13 \arcsin ( x - 1 ) \text {. }


A) 131+2x−x2\frac { 13 } { \sqrt { 1 + 2 x - x ^ { 2 } } }
B) 132x−x2\frac { 13 } { \sqrt { 2 x - x ^ { 2 } } }
C) 131−4x+x2\frac { 13 } { \sqrt { 1 - 4 x + x ^ { 2 } } }
D) 132x+x2\frac { 13 } { \sqrt { 2 x + x ^ { 2 } } }
E) 134x−x2\frac { 13 } { \sqrt { 4 x - x ^ { 2 } } }

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Identify the graph which has the following characteristics. f(0) =−2ft(x) =2,−∞<x<∞\begin{array} { l } f ( 0 ) = - 2 \\f ^ { t } ( x ) = 2 , - \infty < x < \infty\end{array} Identify the graph which has the following characteristics. \begin{array} { l } f ( 0 ) = - 2 \\ f ^ { t } ( x ) = 2 , - \infty < x < \infty \end{array} A) Graph 2 B) Graph 3 C) Graph 1 D) Graph 4 E) none of the above Identify the graph which has the following characteristics. \begin{array} { l } f ( 0 ) = - 2 \\ f ^ { t } ( x ) = 2 , - \infty < x < \infty \end{array} A) Graph 2 B) Graph 3 C) Graph 1 D) Graph 4 E) none of the above


A) Graph 2
B) Graph 3
C) Graph 1
D) Graph 4
E) none of the above

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Determine all values of , (if any) , at which the graph of the function has a horizontal tangent. y(x) =x4−4x+4y ( x ) = x ^ { 4 } - 4 x + 4


A) x=1x = 1
B) x=0x = 0 and x=−1x = - 1
C) x=0x = 0 and x=1x = 1
D) x=0x = 0
E) The graph has no horizontal tangents.

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Complete two iterations of Newton's Method for the function using the initial guess . Round all numerical values in your answer to four decimal places.


A)
nxnf(xn) f′(xn) f(xn) f′(xn) xn−f(xn) f′(xn) 11.31.00000.0068−3.60214.902124.90210.2675−0.26750.00684.8953\begin{array}{|c|c|c|c|c|c|}\hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } & x_{n}-\frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } \\\hline 1 & 1.3 & 1.0000 & 0.0068 & -3.6021 & 4.9021 \\\hline 2 & 4.9021 & 0.2675 & -0.2675 & 0.0068 & 4.8953 \\\hline\end{array}

B)
nxnf(xn) f′(xn) f(xn) f′(xn) xn−f(xn) f′(xn) 11.30.2675−0.9636−0.27761.577621.57760.96360.0068−0.27761.8552\begin{array}{|c|c|c|c|c|c|}\hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } & x_{n}-\frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } \\\hline 1 & 1.3 & 0.2675 & -0.9636 & -0.2776 & 1.5776 \\\hline 2 & 1.5776 & 0.9636 & 0.0068 & -0.2776 & 1.8552 \\\hline\end{array}

C)
nxnf(xn) f′(xn) f(xn) f′(xn) xn−f(xn) f′(xn) 11.30.9636−0.2675−3.60214.902124.90211.00000.0068146.6399−141.7378\begin{array}{|c|c|c|c|c|c|}\hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } & x_{n}-\frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } \\\hline 1 & 1.3 & 0.9636 & -0.2675 & -3.6021 & 4.9021 \\\hline 2 & 4.9021 & 1.0000 & 0.0068 & 146.6399 & -141.7378 \\\hline\end{array}

D)
nxnf(xn) f′(xn) f(xn) f′(xn) xn−f(xn) f′(xn) 11.30.2675−0.9636−0.27761.577621.5776−0.0068−1.00000.00681.5708\begin{array}{|c|c|c|c|c|c|}\hline n & x_{n} & f\left(x_{n}\right) & f^{\prime}\left(x_{n}\right) & \frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } & x_{n}-\frac{f\left(x_{n}\right) }{f^{\prime}\left(x_{n}\right) } \\\hline 1 & 1.3 & 0.2675 & -0.9636 & -0.2776 & 1.5776 \\\hline 2 & 1.5776 & -0.0068 & -1.0000 & 0.0068 & 1.5708 \\\hline\end{array}

E)
nxnf(xn) f′(xn) f(xn) f′(xn) xn−f(xn) f′(xn) 11.30.9636−0.96360.00681.293221.29321.0000−1.0000−3.60214.8953\begin{array}{l}\begin{array} { | c | c | c | c | c | c | } \hline n & x _ { n } & f \left( x _ { n } \right) & f ^ { \prime } \left( x _ { n } \right) & \frac { f \left( x _ { n } \right) } { f ^ { \prime } \left( x _ { n } \right) } & x _ { n } - \frac { f \left( x _ { n } \right) } { f ^ { \prime } \left( x _ { n } \right) } \\\hline 1 & 1.3 & 0.9636 & - 0.9636 & 0.0068 & 1.2932 \\\hline 2 & 1.2932 & 1.0000 & - 1.0000 & - 3.6021 & 4.8953 \\\hline\end{array}\end{array}

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Find the derivative of the function. f(x) =x85−3xf ( x ) = x ^ { 8 } \sqrt { 5 - 3 x }


A) ft(x) =x7(5−51x) 25−3xf ^ { t} ( x ) = \frac { x ^ { 7 } ( 5 - 51 x ) } { 2 \sqrt { 5 - 3 x } }
B) ft(x) =x7(80−3x) 25−3xf ^ { t } ( x ) = \frac { x ^ { 7 } ( 80 - 3 x ) } { 2 \sqrt { 5 - 3 x } }
C) ft(x) =x7(80+51x) 25−3xf ^ { t } ( x ) = \frac { x ^ { 7 } ( 80 + 51 x ) } { 2 \sqrt { 5 - 3 x } }
D) ft(x) =x7(5+3x) 25−3xf ^ { t } ( x ) = \frac { x ^ { 7 } ( 5 + 3 x ) } { 2 \sqrt { 5 - 3 x } }
E) ft(x) =x7(80−51x) 25−3xf ^ { t } ( x ) = \frac { x ^ { 7 } ( 80 - 51 x ) } { 2 \sqrt { 5 - 3 x } }

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Suppose the position function for a free-falling object on a certain planet is given by s(t) =−14t2+v0t+s0s ( t ) = - 14 t ^ { 2 } + v _ { 0 } t + s _ { 0 } . A silver coin is dropped from the top of a building that is 1370 feet tall. Find velocity of the coin at impact. Round your answer to the three decimal places.


A) −286.705ft/sec- 286.705 \mathrm { ft } / \mathrm { sec }
B) −138.492ft/sec- 138.492 \mathrm { ft } / \mathrm { sec }
C) −111.041ft/sec- 111.041 \mathrm { ft } / \mathrm { sec }
D) −276.984ft/sec- 276.984 \mathrm { ft } / \mathrm { sec }
E) −261.984ft/sec- 261.984 \mathrm { ft } / \mathrm { sec }

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ladder feet long is leaning against the wall of a house (see figure) . The base of the ladder is pulled away from the wall at a rate of feet per second. Find the rate at which the angle Between the ladder and the wall of the house is changing when the base of the ladder is feet from The wall. Round your answer to three decimal places. ladder feet long is leaning against the wall of a house (see figure) . The base of the ladder is pulled away from the wall at a rate of feet per second. Find the rate at which the angle Between the ladder and the wall of the house is changing when the base of the ladder is feet from The wall. Round your answer to three decimal places. A) 0.242 \mathrm { rad } / \mathrm { sec } B) 0.190 \mathrm { rad } / \mathrm { sec } C) 2.168 \mathrm { rad } / \mathrm { sec } D) 3.804 \mathrm { rad } / \mathrm { sec } E) 0.278 \mathrm { rad } / \mathrm { sec }


A) 0.242rad/sec0.242 \mathrm { rad } / \mathrm { sec }
B) 0.190rad/sec0.190 \mathrm { rad } / \mathrm { sec }
C) 2.168rad/sec2.168 \mathrm { rad } / \mathrm { sec }
D) 3.804rad/sec3.804 \mathrm { rad } / \mathrm { sec }
E) 0.278rad/sec0.278 \mathrm { rad } / \mathrm { sec }

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man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet above the ground (see figure) . When he is feet from the base of the light, at what rate is the Length of his shadow changing? man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet above the ground (see figure) . When he is feet from the base of the light, at what rate is the Length of his shadow changing? A) \frac { 5 } { 2 } \mathrm { ft } / \mathrm { sec } B) \frac { 65 } { 3 } \mathrm { ft } / \mathrm { sec } C) \frac { 26 } { 3 } \mathrm { ft } / \mathrm { sec } D) \frac { 3 } { 65 } \mathrm { ft } / \mathrm { sec } E) \frac { 1 } { 2 } \mathrm { ft } / \mathrm { sec }


A) 52ft/sec\frac { 5 } { 2 } \mathrm { ft } / \mathrm { sec }
B) 653ft/sec\frac { 65 } { 3 } \mathrm { ft } / \mathrm { sec }
C) 263ft/sec\frac { 26 } { 3 } \mathrm { ft } / \mathrm { sec }
D) 365ft/sec\frac { 3 } { 65 } \mathrm { ft } / \mathrm { sec }
E) 12ft/sec\frac { 1 } { 2 } \mathrm { ft } / \mathrm { sec }

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 Find the derivative of the trigonometric function f(s) =s3tan⁡s. \text { Find the derivative of the trigonometric function } f ( s ) = s ^ { 3 } \tan s \text {. }


A) ft(s) =s3sec⁡2s+3s2tan⁡sf ^ { t } ( s ) = s ^ { 3 } \sec ^ { 2 } s + 3 s ^ { 2 } \tan s
B) ft(s) =s3sec⁡2s+2s2tan⁡sf ^ { t } ( s ) = s ^ { 3 } \sec ^ { 2 } s + 2 s ^ { 2 } \tan s
C) ft(s) =3s2tan⁡s−s2sec⁡2sf ^ {t } ( s ) = 3 s ^ { 2 } \tan s - s ^ { 2 } \sec ^ { 2 } s
D) ft(s) =s2sec⁡2s−3s2tan⁡sf ^ {t } ( s ) = s ^ { 2 } \sec ^ { 2 } s - 3 s ^ { 2 } \tan s
E) ft(s) =s3sec⁡s+3s2tan⁡sf ^ { t } ( s ) = s ^ { 3 } \sec s + 3 s ^ { 2 } \tan s

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A man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet above the ground (see figure) . When he is feet from the base of the light, at what rate is the Tip of his shadow moving? A man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet above the ground (see figure) . When he is feet from the base of the light, at what rate is the Tip of his shadow moving? A) \frac { 1 } { 2 } \mathrm { ft } / \mathrm { sec } B) 50 \mathrm { ft } / \mathrm { sec } C) \frac { 3 } { 50 } \mathrm { ft } / \mathrm { sec } D) \frac { 9 } { 2 } \mathrm { ft } / \mathrm { sec } E) \frac { 50 } { 3 } \mathrm { ft } / \mathrm { sec }


A) 12ft/sec\frac { 1 } { 2 } \mathrm { ft } / \mathrm { sec }
B) 50ft/sec50 \mathrm { ft } / \mathrm { sec }
C) 350ft/sec\frac { 3 } { 50 } \mathrm { ft } / \mathrm { sec }
D) 92ft/sec\frac { 9 } { 2 } \mathrm { ft } / \mathrm { sec }
E) 503ft/sec\frac { 50 } { 3 } \mathrm { ft } / \mathrm { sec }

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 Find an equation of the line that is tangent to the graph of the function f(x) =7x\text { Find an equation of the line that is tangent to the graph of the function } f ( x ) = \frac { 7 } { \sqrt { x } } and parallel to the line 7x+2y−18=07 x + 2 y - 18 = 0 .


A) 7x+y+21=07 x + y + 21 = 0
B) 9x+y−18=09 x + y - 18 = 0
C) 9x+2y+9=09 x + 2 y + 9 = 0
D) 7x+2y−21=07 x + 2 y - 21 = 0
E) 7x+2y−14=07 x + 2 y - 14 = 0

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Find the derivative of the function. f(θ) =75sin⁡22θf ( \theta ) = \frac { 7 } { 5 } \sin ^ { 2 } 2 \theta


A) ft(θ) =−28sin⁡2θcos⁡2θ5f ^ { t } ( \theta ) = - \frac { 28 \sin 2 \theta \cos 2 \theta } { 5 }
B) ft(θ) =28cos⁡2θ5f ^ {t } ( \theta ) = \frac { 28 \cos 2 \theta } { 5 }
C) ft(θ) =28sin⁡2θcos⁡2θ5f ^ { t } ( \theta ) = \frac { 28 \sin 2 \theta \cos 2 \theta } { 5 }
D) ft(θ) =7sin⁡2θcos⁡2θ5f ^ {t } ( \theta ) = \frac { 7 \sin 2 \theta \cos 2 \theta } { 5 }
E) ft(θ) =28sin⁡2θ5f ^ { t } ( \theta ) = \frac { 28 \sin 2 \theta } { 5 }

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 Find an equation to the tangent line to the graph of the function f(x) =tan⁡8x\text { Find an equation to the tangent line to the graph of the function } f ( x ) = \tan ^ { 8 } x at the point (4π5,0.078) \left( \frac { 4 \pi } { 5 } , 0.078 \right) . The coefficients below are given to two decimal places.


A) y=−1.31x+3.36y = - 1.31 x + 3.36
B) y=1.31x+3.36y = 1.31 x + 3.36
C) y=1.06x+3.36y = 1.06 x + 3.36
D) y=−1.31x−3.36y = - 1.31 x - 3.36
E) y=1.06x−3.36y = 1.06 x - 3.36

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 Find dydx by implicit differentiation. \text { Find } \frac { d y } { d x } \text { by implicit differentiation. } x3+8x+x13y−y6=4x ^ { 3 } + 8 x + x ^ { 13 } y - y ^ { 6 } = 4


A) dydx=3x2+8+13x12y6y5−x13\frac { d y } { d x } = \frac { 3 x ^ { 2 } + 8 + 13 x ^ { 12 } y } { 6 y ^ { 5 } - x ^ { 13 } }
B) dydx=3x2+8+13y5y5−13x\frac { d y } { d x } = \frac { 3 x ^ { 2 } + 8 + 13 y } { 5 y ^ { 5 } - 13 x }
C) dydx=3x2+8+13x12y5y5−x13\frac { d y } { d x } = \frac { 3 x ^ { 2 } + 8 + 13 x ^ { 12 } y } { 5 y ^ { 5 } - x ^ { 13 } }
D) dydx=3x2+8+x13y6y5−x13\frac { d y } { d x } = \frac { 3 x ^ { 2 } + 8 + x ^ { 13 } y } { 6 y ^ { 5 } - x ^ { 13 } }
E) dydx=3x2−8+13x12y6y5−x13\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 8 + 13 x ^ { 12 } y } { 6 y ^ { 5 } - x ^ { 13 } }

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The ordering and transportation cost C for the components used in manufacturing a product is C=160(280x2+xx+40) ,x≥1C = 160 \left( \frac { 280 } { x ^ { 2 } } + \frac { x } { x + 40 } \right) , x \geq 1 where CC is measured in thousands of dollars and xx is the order size in hundreds. Find the rate of change of CC with respect to xx for x=24x = 24 . Round your answer to two decimal places.


A) −6.44- 6.44 thousand dollars per hundred
B) 8.048.04 thousand dollars per hundred
C) 3.283.28 thousand dollars per hundred
D) −4.92- 4.92 thousand dollars per hundred
E) −7.96- 7.96 thousand dollars per hundred

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 Use the quotient rule to differentiate the following function f(s) =2ss5+7 and \text { Use the quotient rule to differentiate the following function } f ( s ) = \frac { 2 s } { s ^ { 5 } + 7 } \text { and } evaluate ft(−2) f ^ { t } ( - 2 )


A) ft(−2) =545f ^ { t } ( - 2 ) = \frac { 54 } { 5 }
B) ft(−2) =−543125f ^ { t } ( - 2 ) = - \frac { 54 } { 3125 }
C) ft(−2) =54125f ^ { t } ( - 2 ) = \frac { 54 } { 125 }
D) ft(−2) =−545f ^ {t } ( - 2 ) = - \frac { 54 } { 5 }
E) ft(−2) =−54125f ^ { t } ( - 2 ) = - \frac { 54 } { 125 }

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 Find the derivative of the function f(x) =x6ex. \text { Find the derivative of the function } f ( x ) = x ^ { 6 } e ^ { x } \text {. }


A) 6x5+ex6 x ^ { 5 } + e ^ { x }
B) 6x5+xex6 x ^ { 5 } + x e ^ { x }
C) x5ex(x+6) x ^ { 5 } e ^ { x } ( x + 6 )
D) 6x5ex6 x ^ { 5 } e ^ { x }
E) x5exx ^ { 5 } e ^ { x }

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 Evaluate the derivative of the function f(t) =6t+35t−1 at the point (5,118) . \text { Evaluate the derivative of the function } f ( t ) = \frac { 6 t + 3 } { 5 t - 1 } \text { at the point } \left( 5 , \frac { 11 } { 8 } \right) \text {. }


A) ft(t) =−78f ^ {t } ( t ) = - \frac { 7 } { 8 }
B) ft(t) =7192f ^ { t} ( t ) = \frac { 7 } { 192 }
C) ft(t) =78f ^ { t } ( t ) = \frac { 7 } { 8 }
D) ft(t) =−2126f ^ { t } ( t ) = - \frac { 21 } { 26 }
E) ft(t) =−7192f ^ { t } ( t ) = - \frac { 7 } { 192 }

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