Question 1

Find the derivative  by solving the given implicit equation for y explicitly in terms of x. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

To find the derivative, we need to first solve the given implicit equation for y explicitly in terms of x.

11eb651f_cb62_bfaf_b0a1_2d97cc9e5285_TB8255_11

We can rewrite this equation as:

y^2 + x^2 = 25

Solving for y, we get:

y = ±sqrt(25 - x^2)

Taking the derivative of y with respect to x, we get:

dy/dx = ±(-x/sqrt(25 - x^2))

Simplifying this expression, we get:

dy/dx = ±(-x/|y|)

Since we have both positive and negative values for y, we have to consider both cases. However, we can see that the denominator |y| is always positive, so we can drop the absolute value sign.

Therefore, the derivative is:

dy/dx = -x/y

Plugging in our expression for y, we get:

dy/dx = -x/±sqrt(25 - x^2)

Simplifying the sign, we get:

dy/dx = -x/sqrt(25 - x^2)

This matches choice (C).

Question 2

Find the second derivative  of the function defined implicitly by the equation. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the second derivative  of the...

Question 3

Find the differential of the function.

Accepted Answer

The answer of Find the differential of the function. ...

Question 4

Find  by implicit differentiation. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find  by implicit differentiation. ​ ...

Question 5

Find an equation of the tangent line to the graph of the function f defined by the equation at the indicated point. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

To find the equation of the tangent line, we need to find the derivative of the function and evaluate it at the given point. 
$f(x) = 3x^2 - 2x$ 
$f'(x) = 6x - 2$ 
When x = 1, 
$f'(1) = 6(1) - 2 = 4$ 
So the slope of the tangent line is 4. 
To find the y-intercept of the tangent line, we can use the point-slope form: 
$y - f(1) = m(x - 1) \implies y - 1 = 4(x - 1)$ 
Simplifying, we get: 
$y = 4x - 3$ 
Therefore, the equation of the tangent line at the point (1,1) is y = 4x - 3, which is choice C.

Question 6

Find  by implicit differentiation. ​  ​
A) ​ 
B) 
C) 
D)

Accepted Answer

The answer of Find  by implicit differentiation. ​ ...

Question 7

Find the derivative  by solving the given implicit equation for y explicitly in terms of x. ​  ​
A) 
B) 
C) 2x
D) 2

Accepted Answer

We have the implicit equation:

$11eb651f_cb62_989a_b0a1_63daf280fcee_TB8255_11$,

We need to solve this for $y$ explicitly in terms of $x$.

Rewriting the equation:

$(y^2 - 4)^2 = 4(x^2 - 1)$

Taking the square root of both sides:

$y^2 - 4 = \pm 2\sqrt{x^2 - 1}$

Adding 4 to both sides:

$y^2 = 4 \pm 2\sqrt{x^2 - 1}$

Taking the square root of both sides (remembering that we need to include both the positive and negative square roots):

$y = \pm \sqrt{4 \pm 2\sqrt{x^2-1}}$

Now we can take the derivative of both sides with respect to $x$. Using the chain rule on the right-hand side:

$\frac{dy}{dx} = \pm \frac{1}{2\sqrt{4 \pm 2\sqrt{x^2-1}}} \cdot \frac{d}{dx}\left(4 \pm 2\sqrt{x^2-1}ight)$

$= \pm \frac{1}{2\sqrt{4 \pm 2\sqrt{x^2-1}}} \cdot \left(0 \pm \frac{2x}{\sqrt{x^2-1}}ight)$

$= \pm \frac{x}{\sqrt{(4 \pm 2\sqrt{x^2-1})(x^2-1)}}$

So the final answer is:

$\boxed{	extbf{(B)}\ \frac{x}{\sqrt{(4 \pm 2\sqrt{x^2-1})(x^2-1)}}}$

Question 8

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.? ​
A)  mph
B)  mph
C)  mph
D) 32 mph
E) 24 mph

Accepted Answer

We can use the Pythagorean theorem to find the distance between the two ships at any given time: 
distance^2 = (northward distance)^2 + (eastward distance)^2

At noon (when they both depart), the distance between them is: 
distance = sqrt((0)^2 + (0)^2) = 0

One hour later, the northward distance traveled by Ship A is: 
northward distance = 1 hour * 21 mph = 21 miles

The eastward distance traveled by Ship B is: 
eastward distance = 1 hour * 12 mph = 12 miles

So the distance between the two ships after one hour is: 
distance = sqrt((21)^2 + (12)^2) ≈ 24.186

To find how fast the distance is changing at 1 p.m., we need to find the derivative of distance with respect to time:

distance^2 = (northward distance)^2 + (eastward distance)^2

2(distance)(rate of change of distance) = 2(northward distance)(rate of change of northward distance) + 2(eastward distance)(rate of change of eastward distance)

Substituting the values we know:

2(√(21^2 + 12^2))(rate of change of distance) = 2(21)(0) + 2(12)(-12)

(rate of change of distance) = (-12*12)/(√(21^2 + 12^2)) ≈ -1.977

So the distance between the two ships is decreasing at a rate of approximately 1.977 mph. Answer choice C is the closest option.

Question 9

The demand function for a certain brand of compact disc is  where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Compute the elasticity of demand and determine whether the demand is inelastic, unitary, or elastic when x = 10. ​
A) elastic
B) inelastic
C) unitary

Accepted Answer

To calculate the elasticity of demand, we need to use the formula:

E = (p/x) * (dx/dp)

Where:
- E = elasticity of demand
- p = wholesale unit price
- x = quantity demanded
- dx/dp = derivative of x with respect to p

Taking the derivative of the demand function, we get:

dx/dp = -11e^(-0.0005p)

Substituting p = 10, we get:

dx/dp = -11e^(-0.005) = -10.487

Substituting p = 10 and x = 10, we get:

E = (10/10) * (-10.487) = -10.487

Since the elasticity of demand is negative, the demand is inelastic. Therefore, the best choice is B.

Question 10

A 30-ft ladder leaning against a wall begins to slide. How fast is the top of the ladder sliding down the wall at the instant of time when the bottom of the ladder is 18 ft from the wall and sliding away from the wall at the rate of 3 ft/sec? ​
Hint: Refer to the accompanying figure. By the Pythagorean theorem,  . Find  when x=18 and

a = 30; b = 3
A) 4ft/sec
B)  ft/sec
C)  ft/sec
D) 3 ft/ssec

Accepted Answer

We can use the Pythagorean theorem to find that the height of the ladder (the distance from the top of the ladder to the ground) is 24 ft when the bottom of the ladder is 18 ft from the wall.

Then, we can use the chain rule to find the rate of change of the height with respect to time (dh/dt) when the bottom of the ladder is moving away from the wall at 3 ft/sec:

a^2 + b^2 = c^2
30^2 + 24^2 = c^2
c = 36 ft

c^2 = a^2 + b^2
(36)^2 = (18 + dx)^2 + 24^2

Differentiate both sides with respect to time:

2(36)(dc/dt) = 2(18 + dx)(dx/dt) + 2(24)(dh/dt)

Simplify and plug in known values:

2(36)(-3) = 2(18 + 18)(-3) + 2(24)(dh/dt)
dh/dt = (216/48) ft/sec
dh/dt = 11.25 ft/sec (approx)

Therefore, the top of the ladder is sliding down the wall at a rate of approximately 11.25 ft/sec.

Question 11

Find  by implicit differentiation. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find  by implicit differentiation. ​ ...

Question 12

Let f be the function defined by  . Find the differential of f. ​ 
dy = __________ ​ 
Find the approximate change in y if x changes from 3 to 3.01. ​ 
dy = __________ ​ 
Find the actual change in y if x changes from 3 to 3.01. 
 __________

Accepted Answer

The answer of Let f be the function defined by...

Question 13

Find an equation of the tangent line to the graph of the function f defined by the given equation at the point  . ​  ​
A) 
B) 
C) 
D)

Accepted Answer

We need to find the derivative of f at the given point to get the slope of the tangent line. 
Using the power rule, we get:
f'(x) = 3x^2 - 4x + 1
f'(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) = 3(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)^2 - 4(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) + 1

Simplifying this expression, we get:
f'(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) = 3(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)^2 - 44ebc3a0_8643_5c06_a16e_cdbd0a88167b_TB8255_11 + 1

Now we can use the point-slope form of the equation of a line to find the tangent line. 
y - f(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11) = f'(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)(x - 11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)

Plugging in the values we found, we get:
y - 11eb651f_cb63_5c0a_b0a1_a5c15b7fe69d_TB8255_11 = (3(11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)^2 - 44ebc3a0_8643_5c06_a16e_cdbd0a88167b_TB8255_11 + 1)(x - 11eb651f_cb63_5c06_b0a1_153903464678_TB8255_11)

Simplifying this expression, we get:
y = 3x - 3  (Choice C)

Question 14

Find  by implicit differentiation. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find  by implicit differentiation. ​ ...

Question 15

Find the differential of the function.

Accepted Answer

The answer of Find the differential of the function. ...

Question 16

Find the derivative  by solving the given implicit equation for y explicitly in terms of x. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative  by solving the...

Question 17

Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the equation  . ​ If 25,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 1,000 cartons/week, at what rate is the wholesale price changing?
(Hint: To find the value of p when, solve the supply equation for p when).
A) Increasing at the rate of 3.6 dollars/carton/wk
B) Increasing at the rate of 3.7 dollars/carton/wk
C) Dropping at the rate of 3.7 dollars/carton/wk
D) Dropping at the rate of 3.6 dollars/carton/wk

Accepted Answer

We are given the following information:

p = 15 - 0.0002x (supply equation)
x = 25 - t (where t is the number of weeks)

To find the rate of change of p, we need to take the derivative of p with respect to t:

dp/dt = d/dt (15 - 0.0002x)
dp/dt = -0.0002 * dx/dt

We know that dx/dt = -1,000 because the supply is falling at a rate of 1,000 cartons/week. Substituting this value into the above equation:

dp/dt = -0.0002 * (-1,000)
dp/dt = 0.2

Therefore, the wholesale price is dropping at a rate of 0.2 dollars/carton/week, or 3.7 dollars/carton/wk (rounded to one decimal place). So, the answer is C.

Question 18

Find the differential of the function.

Accepted Answer

The answer of Find the differential of the function. ...

Question 19

Find the differential of the function.

Accepted Answer

The answer of Find the differential of the function. ...

Question 20

In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft. A) 200 ft²/sec B) 400π ft²/sec C) 400 ft²/sec

Accepted Answer

The area of a circle is given by $A = \pi r^2$, where $r$ is the radius. The rate of change of the area with respect to time ($\frac{dA}{dt}$) can be found by differentiating both sides with respect to $t$, giving $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Given $\frac{dr}{dt} = 5$ ft/sec and $r = 40$ ft, substituting these values gives $\frac{dA}{dt} = 2\pi(40)(5) = 400\pi$ ft²/sec.

Quiz 3: Differentiation

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Find the second derivative of the function defined implicitly by the equation.

Find the differential of the function.

Find by implicit differentiation.

Find an equation of the tangent line to the graph of the function f defined by the equation at the indicated point.

Find by implicit differentiation.

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.?

The demand function for a certain brand of compact disc is where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Compute the elasticity of demand and determine whether the demand is inelastic, unitary, or elastic when x = 10.

Find by implicit differentiation.

Let f be the function defined by . Find the differential of f. dy = Find the approximate change in y if x changes from 3 to 3.01. dy = Find the actual change in y if x changes from 3 to 3.01. __

Find an equation of the tangent line to the graph of the function f defined by the given equation at the point .

Find by implicit differentiation.

Find the differential of the function.

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Find the differential of the function.

Find the differential of the function.

In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft.

Quiz 3: Differentiation

Find the derivative by solving the given implicit equation for y explicitly in terms of x. ​ ​

Find the second derivative of the function defined implicitly by the equation. ​ ​

Find the differential of the function.

Find by implicit differentiation. ​ ​

Find an equation of the tangent line to the graph of the function f defined by the equation at the indicated point. ​ ​

Find by implicit differentiation. ​ ​

Find the derivative by solving the given implicit equation for y explicitly in terms of x. ​ ​

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.? ​

The demand function for a certain brand of compact disc is where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Compute the elasticity of demand and determine whether the demand is inelastic, unitary, or elastic when x = 10. ​

Find by implicit differentiation. ​ ​

Let f be the function defined by . Find the differential of f. ​ dy = __________ ​ Find the approximate change in y if x changes from 3 to 3.01. ​ dy = __________ ​ Find the actual change in y if x changes from 3 to 3.01. __________

Find an equation of the tangent line to the graph of the function f defined by the given equation at the point . ​ ​

Find by implicit differentiation. ​ ​

Find the differential of the function.

Find the derivative by solving the given implicit equation for y explicitly in terms of x. ​ ​

Find the differential of the function.

Find the differential of the function.

In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft. ​

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Find the second derivative of the function defined implicitly by the equation.

Find by implicit differentiation.

Find an equation of the tangent line to the graph of the function f defined by the equation at the indicated point.

Find by implicit differentiation.

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.?

The demand function for a certain brand of compact disc is where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Compute the elasticity of demand and determine whether the demand is inelastic, unitary, or elastic when x = 10.

Find by implicit differentiation.

Let f be the function defined by . Find the differential of f. dy = Find the approximate change in y if x changes from 3 to 3.01. dy = Find the actual change in y if x changes from 3 to 3.01. __

Find an equation of the tangent line to the graph of the function f defined by the given equation at the point .

Find by implicit differentiation.

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft.