Question 1

Suppose the total cost in maunufacturing $x$ units of a certain product is $C ( x )$ dollars.
a. What does $C ^ { \prime } ( x )$ measure? Give units.
b. What can you say about the sign of $C ^ { \prime }$ ?
c. Given that $C ^ { \prime } ( 3000 ) = 11$, estimate the additional cost in producing the 3001 st unit of the product.

Accepted Answer

The answer of Suppose the total cost in maunufacturing $x$...

Question 2

The curve with the equation $x ^ { 2 / 3 } + y ^ { 2 / 3 } = 25$ is called an asteroid. Find an equation of the tangent to the curve at the point $( 48 \sqrt { 6 } , 1 )$.

Accepted Answer

The answer of The curve with the equation \(x ^...

Question 3

Find an equation of the tangent line to the given curve at the indicated point.
$$\frac { 1 } { 7 } y ^ { 2 } - x ^ { 3 } - x ^ { 2 } = 0 ; \quad ( 1 , \sqrt { 14 } )$$

Accepted Answer

The answer of Find an equation of the tangent line...

Question 4

The level of nitrogen dioxide present on a certain June day in downtown Megapolis is approximated by
$$A ( t ) = 0.03 t ^ { 3 } ( t - 7 ) ^ { 4 } + 64.8 \quad 0 \leq t \leq 7$$
where $A ( t )$ is measured in pollutant standard index and $t$ is measured in hours with $t = 0$ corresponding to 7 a.m. What is the average level of nitrogen dioxide in the atmosphere from 1 a.m. to 2 p.m. on that day? Round to three decimal places.

Accepted Answer

The answer of The level of nitrogen dioxide present on...

Question 5

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. Show that the curves of the given equations are orthogonal.
$$y - \frac { 7 } { 4 } x = \frac { \pi } { 2 } , \quad x = \frac { 7 } { 4 } \cos y$$

Accepted Answer

The answer of Two curves are said to be orthogonal...

Question 6

The volume of a right circular cone of radius $r$ and height $h$ is $V = \frac { \pi } { 3 } r ^ { 2 } h$. Suppose that the radius and height of the cone are changing with respect to time $t$.
a. Find a relationship between $\frac { d V } { d t } , \frac { d r } { d t }$, and $\frac { d h } { d t }$.
b. At a certain instant of time, the radius and height of the cone are 12 in. and 13 in. and are increasing at the rate of $0.2 \mathrm { in } . / \mathrm { sec }$ and $0.5 \mathrm { in } . / \mathrm { sec }$, respectively. How fast is the volume of the cone increasing?

Accepted Answer

The answer of The volume of a right circular cone...

Question 7

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

Accepted Answer

The answer of Find the derivative of the function.
\[f (...

Question 8

$$\text { Sketch the graph of the derivative } f ^ { \prime } \text { of the function } f \text { whose graph is given. }$$

Accepted Answer

The answer of \[\text { Sketch the graph of the...

Question 9

Identify the "inside function" $u = f ( x )$ and the "outside function" $y = g ( u )$. Then find $d y / d x$ using the Chain Rule.
$$y = \sqrt { x ^ { 2 } - 2 }$$

Accepted Answer

The answer of Identify the "inside function" \(u = f...

Question 10

The graph shows the percentage of households in a certain city watching television during a $24 - h r$ period on a weekday ( $t = 0$ corresponds to 6 a.m.). By computing the slope of the respective tangent line, estimate the rate of change of the percentage of households watching television at a-12 p.m.
Note that $d y = 0.03$

Accepted Answer

The answer of The graph shows the percentage of households...

Question 11

Find the derivative of the function.
$$f ( x ) = x \sin ^ { 8 } x$$

Accepted Answer

The answer of Find the derivative of the function.
\[f (...

Question 12

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is $20 \mathrm { ft }$ and is increasing at the rate of $\frac { 1 } { 6 } \mathrm { ft } / \mathrm { sec }$. Round to the nearest tenth if necessary.

Accepted Answer

The answer of In calm waters, the oil spilling from...

Question 13

Differentiate.
$$y = \frac { \sin x } { 3 + \cos x }$$

Accepted Answer

The answer of Differentiate.
\[y = \frac { \sin x }...

Question 14

$s ( t )$ 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.
$$s ( t ) = \frac { 4 t } { t ^ { 2 } + 1 } ; \quad t = 3$$

Accepted Answer

The answer of $s ( t )$ is the position...

Question 15

Let $f ( x ) = x \left| x ^ { 3 } \right|$.
a. Sketch the graph of $f$.
b. For what values of $x$ is $f$ differentiable?
c. Find a formula for $f ^ { \prime } ( x )$.

Accepted Answer

The answer of Let \(f ( x ) = x...

Question 16

Find the limit.
$$\lim _ { \theta \rightarrow 0 } 4 \frac { \sin ( \sin 4 \theta ) } { \sec 4 \theta }$$

Accepted Answer

The answer of Find the limit.
\[\lim _ { \theta \rightarrow...

Question 17

Suppose that $f$ and $g$ are functions that are differentiable at $x = 1$ and that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - 3 , g$ $( 1 ) = 2$, and $g ^ { \prime } ( 1 ) = 5$. Find $h ^ { \prime } ( 1 )$.
$$h ( x ) = \frac { x f ( x ) } { x + g ( x ) }$$

Accepted Answer

The answer of Suppose that $f$ and $g$ are functions...

Question 18

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is $20 \mathrm { ft }$ and is increasing at the rate of $\frac { 1 } { 6 } \mathrm { ft } / \mathrm { sec }$. Round to the nearest tenth if necessary.

Accepted Answer

The answer of In calm waters, the oil spilling from...

Question 19

The sides of a square baseball diamond are $90 \mathrm { ft }$ long. When a player who is between the second and third base is $30 \mathrm { ft }$ from second base and heading toward third base at a speed of $24 \mathrm { ft } / \mathrm { sec }$, how fast is the distance between the player and home plate changing? Round to two decimal places.

Accepted Answer

The answer of The sides of a square baseball diamond...

Quiz 1: Functions and Models

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

$\text { Sketch the graph of the derivative } f ^ { \prime } \text { of the function } f \text { whose graph is given. }$ $\text { Sketch the graph of the derivative } f ^ { \prime } \text { of the function } f \text { whose graph is given. }$

Identify the "inside function" $u = f ( x )$ and the "outside function" $y = g ( u )$ . Then find $d y / d x$ using the Chain Rule. $y = \sqrt { x ^ { 2 } - 2 }$

Find the derivative of the function. $f ( x ) = x \sin ^ { 8 } x$

Differentiate. $y = \frac { \sin x } { 3 + \cos x }$

$s ( t )$ 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. $s ( t ) = \frac { 4 t } { t ^ { 2 } + 1 } ; \quad t = 3$

Let $f ( x ) = x \left| x ^ { 3 } \right|$ . a. Sketch the graph of $f$ . b. For what values of $x$ is $f$ differentiable? c. Find a formula for $f ^ { \prime } ( x )$ .

Find the limit. $\lim _ { \theta \rightarrow 0 } 4 \frac { \sin ( \sin 4 \theta ) } { \sec 4 \theta }$

Suppose that $f$ and $g$ are functions that are differentiable at $x = 1$ and that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - 3 , g$ $( 1 ) = 2$ , and $g ^ { \prime } ( 1 ) = 5$ . Find $h ^ { \prime } ( 1 )$ . $h ( x ) = \frac { x f ( x ) } { x + g ( x ) }$

Quiz 1: Functions and Models

The curve with the equation x2/3+y2/3=25x ^ { 2 / 3 } + y ^ { 2 / 3 } = 25x2/3+y2/3=25 is called an asteroid. Find an equation of the tangent to the curve at the point (486,1)( 48 \sqrt { 6 } , 1 )(486​,1) .

Find an equation of the tangent line to the given curve at the indicated point. 17y2−x3−x2=0;(1,14)\frac { 1 } { 7 } y ^ { 2 } - x ^ { 3 } - x ^ { 2 } = 0 ; \quad ( 1 , \sqrt { 14 } )71​y2−x3−x2=0;(1,14​)

Find the derivative of the function. f(x)=−x2+x+2f ( x ) = - x ^ { 2 } + x + 2f(x)=−x2+x+2

Sketch the graph of the derivative f′ of the function f whose graph is given. \text { Sketch the graph of the derivative } f ^ { \prime } \text { of the function } f \text { whose graph is given. } Sketch the graph of the derivative f′ of the function f whose graph is given.

Identify the "inside function" u=f(x)u = f ( x )u=f(x) and the "outside function" y=g(u)y = g ( u )y=g(u) . Then find dy/dxd y / d xdy/dx using the Chain Rule. y=x2−2y = \sqrt { x ^ { 2 } - 2 }y=x2−2​

Find the derivative of the function. f(x)=xsin⁡8xf ( x ) = x \sin ^ { 8 } xf(x)=xsin8x

Differentiate. y=sin⁡x3+cos⁡xy = \frac { \sin x } { 3 + \cos x }y=3+cosxsinx​

Let f(x)=x∣x3∣f ( x ) = x \left| x ^ { 3 } \right|f(x)=x​x3​ . a. Sketch the graph of fff . b. For what values of xxx is fff differentiable? c. Find a formula for f′(x)f ^ { \prime } ( x )f′(x) .

Find the limit. lim⁡θ→04sin⁡(sin⁡4θ)sec⁡4θ\lim _ { \theta \rightarrow 0 } 4 \frac { \sin ( \sin 4 \theta ) } { \sec 4 \theta }limθ→0​4sec4θsin(sin4θ)​