Exam 2: Limits and Derivatives

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 $\frac { \pi } { 3 }$ .

A point moves along the curve $3 y + y ^ { 2 } - 8 x = 2$ . When the point is at $\left( - \frac { 1 } { 2 } , - 1 \right)$ , its $x$ -coordinate is increasing at the rate of 3 units per second. How fast is its $y$ -coordinate changing at that instant of time?

Find equations of the tangent lines to the curve $y = \frac { x - 10 } { x + 10 }$ that are parallel to the line $x - y = 10$ .

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 $\frac { \pi } { 3 }$ . Select the correct answer.

Find the differential of the function at the indicated number. Select the correct answer. $f ( x ) = \sqrt { x ^ { 2 } + 7 } ; \quad x = 3$

A car leaves an intersection traveling west. Its position $4 \mathrm { sec }$ later is $26 \mathrm { ft }$ from the intersection. At the same time, another car leaves the same intersection heading north so that its position $4 \mathrm { sec }$ later is $26 \mathrm { ft }$ from the intersection. If the speeds of the cars at that instant of time are $12 \mathrm { ft } / \mathrm { sec }$ and 10 $\mathrm { ft } / \mathrm { sec }$ , respectively, find the rate at which the distance between the two cars is changing. Round to the nearest tenth if necessary.

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 $3 \mathrm {~m}$ .

A turkey is removed from the oven when its temperature reaches $175 ^ { \circ } \mathrm { F }$ and is placed on a table in a room where the temperature is $70 ^ { \circ } \mathrm { F }$ . After 10 minutes the temperature of the turkey is 161 ${ } ^ { \circ } \mathrm { F }$ and after 20 minutes it is $149 ^ { \circ } \mathrm { F }$ . Use a linear approximation to predict the temperature of the turkey after 30 minutes.

A turkey is removed from the oven when its temperature reaches $175 ^ { \circ } \mathrm { F }$ and is placed on a table in a room where the temperature is $70 ^ { \circ } \mathrm { F }$ . After 10 minutes the temperature of the turkey is $160 ^ { \circ } \mathrm { F }$ and after 20 minutes it is $150 ^ { \circ } \mathrm { F }$ . Use a linear approximation to predict the temperature of the turkey after 40 minutes. Select the correct answer.

Let $C ( t )$ be the total value of US currency (coins and banknotes) in circulation at time. The table gives values of this function from 1980 to 2000 , as of September 30 , in billions of dollars. Estimate the value of $C ( 1990 )$ . 1980 1985 1990 1995 2000 () 129.9 176.3 275.9 405.3 568.6 Answers are in billions of dollars per year. Round your answer to two decimal places.

The equation of motion is given for a particle, where $s$ is in meters and $t$ is in seconds. Find the acceleration after $2.5$ seconds. $s = \sin 2 \pi t$

In an adiabatic process (one in which no heat transfer takes place), the pressure $P$ and volume $V$ of an ideal gas such as oxygen satisfy the equation $P ^ { 5 } V ^ { 7 } = C$ where $C$ is a constant. Suppose that at a certain instant of time, the volume of the gas is $2 \mathrm {~L}$ , the pressure is $100 \mathrm { kPa }$ , and the pressure is decreasing at the rate of $5 \mathrm { kPa } / \mathrm { sec }$ . Find the rate at which the volume is changing.

Find the derivative of the function. $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( \frac { 9 x - 1 } { 7 x + 1 } \right)$

Use the linear approximation of the function $f ( x ) = \sqrt { 9 - x }$ at $a = 0$ to approximate the number $\sqrt { 9.08 }$ .

Find the second derivative of the function. Select the correct answer. $f ( x ) = x \left( 3 x ^ { 2 } - 1 \right) ^ { 4 }$

Find the point(s) on the graph of $f$ where the tangent line is horizontal. $f ( x ) = x ^ { 2 } e ^ { - x }$

$s ( t )$ is the position of a body moving along a coordinate line; $s ( t )$ is measured in feet and $t$ in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s ( t ) = \frac { 4 t } { t ^ { 2 } + 1 } ; \quad t = 3$

Find the derivative of the function. $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( \frac { 9 x - 1 } { 7 x + 1 } \right)$

Differentiate the function. $B ( y ) = c y ^ { - 4 }$