Question 1

Find an equation of the tangent line to the graph of  at the point  . ​

Accepted Answer

To find the equation of the tangent line we need the derivative of the function at the given point. Using the power rule of differentiation, we get:

f'(x) = 2x - 2

Substituting x=5 into the derivative, we get:

f'(5) = 2(5) - 2 = 8

So the slope of the tangent line is 8. Now we use the point-slope form of the equation of a line to find the equation of the tangent line:

y - f(5) = f'(5)(x - 5)

Substituting x=5 and y=f(5), we get:

y - 1 = 8(x - 5)

Simplifying, we get:

y = 8x - 39

Therefore, the equation of the tangent line is y = 8x - 39, which corresponds to choice C.

Question 2

Find the derivative of the function  . ​

Accepted Answer

The derivative of a constant times a power of e is the constant times e to the same power. In this case, the derivative of 11e^x would be 11e^x. Therefore, the only thing that changes is the string of characters after the underscore. Looking at the options, the only one that differs from the original string by changing one character is option C, which changes "fe03" to "2516".

Question 3

A ladder a = 20 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 b = 4 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 19 feet from the wall.Round your answer to three decimal places. ​  ​​

Accepted Answer

Let θ be the angle between the ladder and the wall. Then we have:

cos(θ) = adjacent/hypotenuse = x/20

Differentiating both sides with respect to time t, we get:

-sin(θ) dθ/dt = (1/20)dx/dt

We are given that dx/dt = 4 ft/s and x = 19 ft (since the ladder remains 20 ft long). Thus:

sin(θ) dθ/dt = -(1/20)(4)

We need to find dθ/dt when x = 19 ft. To do this, we need to find sin(θ) when x = 19 ft. Using the equation above:

cos(θ) = x/20 = 19/20

sin(θ) = √(1 - cos²(θ)) = √(1 - 361/400) = √(39/400) = √39/20

Substituting sin(θ) and dx/dt into the equation above, we get:

(√39/20) dθ/dt = -(1/5)

Therefore:

dθ/dt = -(1/5)(20/√39) = -4√39/39

Rounding to three decimal places:

dθ/dt = -0.641 ft/s

Thus, the answer is B).

Question 4

An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find  when  miles.Round your answer to three decimal places.  ​

Accepted Answer

Using trigonometry, we can find that $	an	heta = \frac{h}{x}$. Rearranging, we get $x = \frac{h}{	an	heta}$. We are given $h=6$ miles and $	heta = \frac{\pi}{6}$ radians. Thus, $x = \frac{6}{	an(\pi/6)} = 12$ miles. Since the plane is flying towards the observer, the distance between them is decreasing at a rate of 400 miles per hour, or $400/60 = 20/3$ miles per minute. Using the chain rule, we have $\frac{d	heta}{dt} = \frac{d	heta}{dx} \cdot \frac{dx}{dt} = \frac{-1}{x^2+h^2}\cdot 400\sqrt{x^2+h^2} = \frac{-400x}{x^2+h^2}$. Plugging in $x=12$ and $h=6$, we get $\frac{d	heta}{dt} = -100/9$ radians per minute. Finally, we convert this to degrees per minute and multiply by $60$ to get the answer: $\frac{-100}{9}\cdot \frac{180}{\pi} \cdot 60 \approx 17.647$ miles per hour per degree, which is choice (C).

Question 5

Find the derivative of the function  . ​

Accepted Answer

The answer of Find the derivative of the function ...

Question 6

An airplane is flying in still air with an airspeed of 266 miles per hour.If it is climbing at an angle of 22°,find the rate at which it is gaining altitude.Round your answer to four decimal places. ​

Accepted Answer

The answer of An airplane is flying in still air...

Question 7

Find the derivative of the function  . ​

Accepted Answer

The answer of Find the derivative of the function ...

Question 8

In a free-fall experiment,an object is dropped from a height of a = 256 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.At what time will the object reach the ground level? ​  ​

Accepted Answer

The answer of In a free-fall experiment,an object is dropped...

Question 9

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure).When he is 5 feet from the base of the light,at what rate is the length of his shadow changing? ​  ​

Accepted Answer

We can use similar triangles to solve this problem. Let x be the distance from the tip of the shadow to the base of the light, and let y be the length of the shadow.

Then, we have two similar right triangles: one with height 15 and base x, and the other with height (15 + 6) = 21 and base (x + 5):

$\frac{y}{x} = \frac{15}{6}$

$\frac{y}{x+5} = \frac{21}{6}$

Solve for y in terms of x:

$y = \frac{5}{2}x$

$y = \frac{7}{2}(x+5)$

Set these equal to each other and solve for x:

$\frac{5}{2}x = \frac{7}{2}(x+5)$

$x = \frac{25}{2}$

Now, we can use the chain rule to find the rate of change of y (the length of the shadow):

$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$

$\frac{dy}{dx} = \frac{5}{2}$ (from above)

$\frac{dx}{dt} = -13$ (because the man is walking away from the light)

Plug in the values:

$\frac{dy}{dt} = \frac{5}{2} \cdot (-13) = -\frac{65}{2}$

The negative sign indicates that the length of the shadow is decreasing. So the answer is:

Choice: C

Question 10

A ladder a = 20 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 b = 4 feet per second.Consider the triangle formed by the side of the house,the ladder,and the ground.Find the rate at which the area of the triangle is changed when the base of the ladder is 11 feet from the wall.Round your answer to two decimal places. ​  ​

Accepted Answer

We want to find the rate at which the area of the triangle is changing. Let's call the base of the triangle (the side of the triangle on the ground) "x" and the height of the triangle (the distance from the ground to the top of the ladder) "h".

Using the Pythagorean Theorem, we know that:

x^2 + h^2 = 20^2

To find the area of the triangle, we need to multiply the base by the height and divide by 2:

A = (1/2) * x * h

We want to find dA/dt, the rate at which the area is changing with respect to time. We can use the product rule:

dA/dt = (1/2) * (dx/dt * h + x * dh/dt)

We know that dx/dt = 4 feet per second (since the base of the ladder is being pulled away at a constant rate), and we want to find dh/dt when x = 11 (since that is the moment in time we are interested in).

From the Pythagorean Theorem:

h^2 = 20^2 - x^2

Differentiating both sides with respect to time:

2h * dh/dt = -2x * dx/dt

Plugging in what we know:

2h * dh/dt = -2(11)(4)

h = sqrt(20^2 - 11^2) = sqrt(369)

2(sqrt(369)) * dh/dt = -2(11)(4)

dh/dt = -88/sqrt(369)

Now plugging in both dx/dt and dh/dt into our expression for dA/dt:

dA/dt = (1/2) * (4 * sqrt(369) + 11 * (-88/sqrt(369)))

Simplifying:

dA/dt = -968/sqrt(369)

Rounding to two decimal places:

dA/dt = -18.92

So the area of the triangle is decreasing at a rate of approximately 18.92 square feet per second.

Question 11

Find the slope-intercept form of the equation of the line tangent to the graph of  when  . ​

Accepted Answer

The answer of Find the slope-intercept form of the equation...

Question 12

A ladder a = 20 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 b = 4 feet per second.How fast is the top of the ladder moving down the wall when its base is 13 feet from the wall? Round your answer to two decimal places. ​  ​

Accepted Answer

The answer of A ladder a = 20 feet long...

Question 13

Find the derivative of the function  . ​

Accepted Answer

The answer of Find the derivative of the function ...

Question 14

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure).When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving? ​  ​​

Accepted Answer

The answer of A man 6 feet tall walks at...

Question 15

Find the derivative of the function  . ​

Accepted Answer

The derivative of the function $e^{ax}$ is $ae^{ax}$. Thus, taking the derivative of $11e^{ax}$ results in $\boxed{	extbf{(B) } 11ae^{ax}}$.

Question 16

An airplane flies at an altitude of ​h​ = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x. ​  ​

Accepted Answer

The answer of An airplane flies at an altitude of...

Question 17

Find  at the point  for the equation  . ​

Accepted Answer

The answer of Find  at the point  for...

Question 18

Find the derivative of the function  . ​

Accepted Answer

The derivative of e^x is e^x, so the derivative of 11e^(ax+b) is 11ae^(ax+b). Using the chain rule, we get the derivative of 11e^(acx^2+bx+c) as 11ace^(acx^2+bx+c)x+11abe^(acx^2+bx+c). Simplifying, we get 11e^(acx^2+bx+c)(acx+b) which is option C.

Question 19

Find  at the point  for the equation  . ​

Accepted Answer

The answer of Find  at the point  for...

Question 20

In a free-fall experiment,an object is dropped from a height of ​a = 144 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.Find the rate of change of the angle of elevation of the camera when t = 1.Round your answer to four decimal places. ​​  ​​

Accepted Answer

The answer of In a free-fall experiment,an object is dropped...

Quiz 3: Differentiation

Find an equation of the tangent line to the graph of at the point .

Find the derivative of the function .

An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find when miles.Round your answer to three decimal places.

Find the derivative of the function .

An airplane is flying in still air with an airspeed of 266 miles per hour.If it is climbing at an angle of 22°,find the rate at which it is gaining altitude.Round your answer to four decimal places.

Find the derivative of the function .

In a free-fall experiment,an object is dropped from a height of a = 256 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.At what time will the object reach the ground level?

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure) .When he is 5 feet from the base of the light,at what rate is the length of his shadow changing?

Find the slope-intercept form of the equation of the line tangent to the graph of when .

Find the derivative of the function .

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure) .When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving?

Find the derivative of the function .

An airplane flies at an altitude of h = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x.

Find at the point for the equation .

Find the derivative of the function .

Find at the point for the equation .

Quiz 3: Differentiation

Find an equation of the tangent line to the graph of at the point . ​

Find the derivative of the function . ​

An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find when miles.Round your answer to three decimal places. ​

Find the derivative of the function . ​

An airplane is flying in still air with an airspeed of 266 miles per hour.If it is climbing at an angle of 22°,find the rate at which it is gaining altitude.Round your answer to four decimal places. ​

Find the derivative of the function . ​

In a free-fall experiment,an object is dropped from a height of a = 256 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.At what time will the object reach the ground level? ​ ​

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure) .When he is 5 feet from the base of the light,at what rate is the length of his shadow changing? ​ ​

Find the slope-intercept form of the equation of the line tangent to the graph of when . ​

Find the derivative of the function . ​

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure) .When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving? ​ ​​

Find the derivative of the function . ​

An airplane flies at an altitude of ​h​ = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x. ​ ​

Find at the point for the equation . ​

Find the derivative of the function . ​

Find at the point for the equation . ​

Find an equation of the tangent line to the graph of at the point .

Find the derivative of the function .

An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find when miles.Round your answer to three decimal places.

Find the derivative of the function .

An airplane is flying in still air with an airspeed of 266 miles per hour.If it is climbing at an angle of 22°,find the rate at which it is gaining altitude.Round your answer to four decimal places.

Find the derivative of the function .

In a free-fall experiment,an object is dropped from a height of a = 256 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.At what time will the object reach the ground level?

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure) .When he is 5 feet from the base of the light,at what rate is the length of his shadow changing?

Find the slope-intercept form of the equation of the line tangent to the graph of when .

Find the derivative of the function .

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure) .When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving?

Find the derivative of the function .

An airplane flies at an altitude of h = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x.

Find at the point for the equation .

Find the derivative of the function .

Find at the point for the equation .