Exam 2: Limits and Derivatives

Find an equation of the tangent line to the curve $120 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2312 \left( x ^ { 2 } - y ^ { 2 } \right)$ at the point $( 4,1 )$ . Select the correct answer.

Find the equation of the tangent to the curve at the given point. $y = \sqrt { 16 + 4 \sin x } , ( 0,4 )$

Find the derivative of the function. $f ( x ) = - x ^ { 2 } + x + 2$

If $f ( t ) = \sqrt { 9 t + 1 }$ , find $f ^ { \prime \prime } ( 5 )$

The mass of the part of a metal rod that lies between its left end and a point $x$ meters to the right is $S = 4 x ^ { 2 }$ Find the linear density when $x$ is $3 \mathrm {~m}$ .

Two sides of a triangle are $2 \mathrm {~m}$ and $3 \mathrm {~m}$ in length and the angle between them is increasing at a rate of $0.06 \mathrm { rad } / \mathrm { s }$ . Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\frac { \pi } { 3 }$ .

Use the Quotient Rule to find the derivative of the function. $P ( t ) = \frac { 1 - t } { 7 - 8 t }$

Differentiate. $K ( x ) = \left( 3 x ^ { 5 } + 1 \right) \left( x ^ { 6 } - 4 x \right)$

A spherical balloon is being inflated. Find the rate of increase of the surface area $S = 4 \pi r ^ { 2 }$ with respect to the radius $r$ when $r = 1 \mathrm { ft }$ .

If $f$ is the focal length of a convex lens and an object is placed at a distance $v$ from the lens, then its image will be at a distance $u$ from the lens, where $f , v$ , and $u$ are related by the lens equation $\frac { 1 } { f } = \frac { 1 } { v } + \frac { 1 } { u }$ Find the rate of change of $v$ with respect to $u$ .

$s ( t )$ is the position of a body moving along a coordinate line, where $t \geq 0$ , and $s ( t )$ is measured in feet and $t$ in seconds. $s ( t ) = - 3 + 2 t - t ^ { 2 }$ a. Determine the time(s) and the position(s) when the body is stationary. b. When is the body moving in the positive direction? In the negative direction? c. Sketch a schematic showing the position of the body at any time $t$ .

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. $y = \sin x y ^ { 6 } ; \quad \left( \frac { \pi } { 2 } , 1 \right)$

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0017 \mathrm {~cm}$ thick to a hemispherical dome with diameter $70 \mathrm {~m}$ .

Water flows from a tank of constant cross-sectional area $50 \mathrm { At } ^ { 2 }$ through an orifice of constant cross-sectional area $\frac { 1 } { 4 } \mathrm { ft } ^ { 2 }$ located at the bottom of the tank. Initially, the height of the water in the tank was $20 \mathrm { ft }$ , and $t$ sec later it was given by the equation $2 \sqrt { h } + \frac { 1 } { 25 } t - 2 \sqrt { 20 } = 0 \quad 0 \leq t \leq 50 \sqrt { 20 }$ How fast was the height of the water decreasing when its height was $2 \mathrm { ft }$ ?

The cost (in dollars) of producing $x$ units of a certain commodity is $C ( x ) = 4,280 + 13 x + 0.03 x ^ { 2 }$ Find the average rate of change with respect to $x$ when the production level is changed from $x = 102$ to $x = 122$ .

If a snowball melts so that its surface area decreases at a rate of $4 \mathrm {~cm} ^ { 2 } / \mathrm { min }$ , find the rate at which the diameter decreases when the diameter is $37 \mathrm {~cm}$ .

Find the derivative of the function. $f ( x ) = ( 4 x + 9 ) ^ { 9 }$

Find the derivative of the function. $y = 3 \cos ^ { - 1 } \left( \sin ^ { - 1 } t \right)$

Use the Product Rule to find the derivative of the function. Select the correct answer. $f ( x ) = ( 4 x + 5 ) \left( x ^ { 2 } - 8 \right)$