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AP Calculus AB : AP Calculus AB

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Example Questions

Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems

A spherical balloon is increasing in volume at a constant rate of 0.1 cm^3/min . At a radius of 3 cm, what is the rate of change of the circumference of the balloon?

Possible Answers:

$\frac{1}{180} cm/min$

$\frac{1}{90\pi} cm/min$

$\frac{1}{360} cm/min$

$\frac{1}{360\pi} cm/min$

$\frac{1}{90} cm/min$

Correct answer:

$\frac{1}{180} cm/min$

Explanation:

To determine the rate of change of the circumference at a given radius, we must relate the circumference rate of change to the rate of change we know - that of the volume.

Starting with the equation for the volume of the spherical balloon,

$V=\frac{4}{3}\pi r^3$

we take the derivative of the function with respect to time, giving us the rate of change of the volume:

$\frac{\mathrm{d} V}{\mathrm{d} t}=4\pi r^2 \frac{\mathrm{d} r}{\mathrm{d} t}$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$

The chain rule was used when taking the derivative of the radius with respect to time, because we know that it is a function of time.

Solving for $\frac{\mathrm{d} r}{\mathrm{d} t}$ using our known $\frac{\mathrm{d} V}{\mathrm{d} t}$ at the given radius, we get

$\frac{\mathrm{d} r}{\mathrm{d} t}=\frac{1}{360\pi} cm/min$

Now, we use this rate of change and apply it to the rate of change of the circumference, which we get by taking the derivative of the circumference with respect to time:

$c=2\pi r$

$\frac{\mathrm{d} c}{\mathrm{d} t}=2\pi \frac{\mathrm{d} r}{\mathrm{d} t}$

Solving for the rate of change of the circumference by plugging in the known rate of change of the radius, we get

$\frac{\mathrm{d} c}{\mathrm{d} t}=2\pi (\frac{1}{360 \pi})=\frac{1}{180} cm/min$

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Example Question #1 : Modeling Rates Of Change, Including Related Rates Problems

Determine the rate of change of the angle opposite the base of a right triangle -whose length is increasing at a rate of 1 inch per minute, and whose height is a constant 2 inches - when the area of the triangle is 2 square inches.

Possible Answers:

$\frac{1}{4\sqrt{2}}$ radians per minute

$\frac{1}{4}$ radians per minute

$\frac{1}{2\sqrt{2}}$ radians per minute

$\frac{1}{\sqrt{2}}$ radians per minute

Correct answer:

$\frac{1}{\sqrt{2}}$ radians per minute

Explanation:

To determine the rate of the change of the angle opposite to the base of the given right triangle, we must relate it to the rate of change of the base of the triangle when the triangle is a certain area.

First we must determine the length of the base of the right triangle at the given area:

$A=\frac{bh}{2}$

4=b(2)

b=2

Now, we must find something that relates the angle opposite of the base to the length of the base and height - the tangent of the angle:

$\tan \theta=\frac{b}{h}$

To find the rate of change of the angle, we take the derivative of both sides with respect to time, keeping in mind that the base of the triangle is dependent on time, while the height is constant:

$\sec^2(\theta)\frac{\mathrm{d} \theta}{\mathrm{d} t}=\frac{2}{2}\frac{\mathrm{d} b}{\mathrm{d} t}$

We know the rate of change of the base, and we can find the angle from the sides of the triangle:

$\tan(\theta)=1$

$\theta=\frac{\pi}{4}$

Plugging this and the other known information in and solving for the rate of change of the angle adjacent to the base, we get

$\sec^2(\frac{\pi}{4})\frac{\mathrm{d} \theta}{\mathrm{d} t}=1$

$\frac{\mathrm{d} \theta}{\mathrm{d} t}=\frac{1}{\sqrt{2}}$ radians per minute

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Example Question #2 : Applications Of Derivatives

The position of a car is given by the equation

$f(t) = sin(\pi t) + 3t^{2} - 10t + 7$ .

Find the car's acceleration when t = 4 .

Possible Answers:

$\pi$ + 14

$-\pi^{3}$

Correct answer:

Explanation:

To find the car's acceleration, take the SECOND derivative of f(t) .

$f'(t) = \pi cos(\pi t) + 6t - 10$ , and

$f''(t) = -\pi ^{2}sin(\pi t) + 6$ .

The car's position at t = 4 is then given by:

$f''(4) = -\pi ^{2}sin(\pi *4) + 6 = 6$

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Example Question #1 : Applications Of Derivatives

A point on a circle of radius 1 unit is orbiting counter-clockwise around the circle's center. It makes one full orbit every 8 seconds. How fast is the coordinate changing when the line segment from the origin to the point, (x,y) , forms an angle of $\frac{\pi}{4}$ radians above the positive x axis?

Possible Answers:

$\frac{\sqrt{2}\pi}{8}$

$\frac{-\sqrt{2}\pi}{8}$

$\frac{1}{2}$

$-\frac{\sqrt{3}\pi}{8}$

$\frac{-\sqrt{2}}{2}$

Correct answer:

$\frac{-\sqrt{2}\pi}{8}$

Explanation:

This is a related rates problem.

Since the problem gives the time for one orbit, we can find the angular speed of the point. The angular speed is simply how many radians the particle travels in one second. We find this by dividing the amount of radians in one revolution, $2\pi$ , by the time it takes to travel one revolution, 8 seconds.

$\frac{2\pi}{8} = \frac{\pi}{4}$

This gives us the change in the angle with respect to time, $\frac{\mathrm{d} \theta}{\mathrm{d} t} = \frac{\pi}{4}$ .

Now we need to relate the position to the angle, $\theta$ . Recall that

$\cos \theta = \frac{x}{r}$ , where r is the radius.

Since the radius is given as 1 unit, we can write this equation as

$x = \cos \theta$ .

Now we take the derivative of both sides with respect to time, using implicit differentiation. Remember that we use the chain rule for any variable that is not . This gives,

$\frac{\mathrm{d} x}{\mathrm{d} t} = -\sin\theta \frac{\mathrm{d} \theta}{\mathrm{d} t}$ .

Now we have a formula that relates the horizontal speed of the particle at a instant in time, $\frac{\mathrm{d} x}{\mathrm{d} t}$ , to the angle above the positive x axis and angular speed at that same instant. We are told to find how fast the x coordinate is changing when the angle, $\theta$ is $\frac{\pi}{4}$ radians above the positive x axis. So we will plug in $\frac{\pi}{4}$ for $\theta$ . However we also need to know $\frac{\mathrm{d} \theta}{\mathrm{d} t}$ . Fortunately, we already found it. It is the angular speed, $\frac{\pi}{4}$ radians/second.

Plugging this information in, we get

$\frac{\mathrm{d} x}{\mathrm{d} t}= - \sin{(\frac{\pi}{4})} (\frac{\pi}{4})$

$\frac{\mathrm{d} x}{\mathrm{d} t} = -(\frac{\sqrt{2}}{2})(\frac{\pi}{4})$

$\frac{\mathrm{d} x}{\mathrm{d} t}= \frac{-\sqrt{2}\pi}{8}$

This is the answer. The negative makes sense, because the point is traveling counter-clockwise. Hence, it is moving left when the angle is $\frac{\pi}{4}$ radians.

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Example Question #2 : Applications Of Derivatives

A man is standing on the top of a 10 ft long ladder that is leaning against the side of a building, when the bottom of the ladder begins to slide out from under it. How fast is the man standing on the top of the ladder falling when the bottom of the ladder is 6 ft from the building and is sliding at 2ft/sec?

Possible Answers:

$-1/4 \text{ feet/sec}$

$-3/2 \text{ feet/sec}$

$32 \text{ feet/sec}$

$3/4 \text{ feet/sec}$

Correct answer:

$-3/2 \text{ feet/sec}$

Explanation:

This is a related rates problem. The ladder leaning against the side of a building forms a right triangle, with the 10ft ladder as its hypotenuse. The Pythagorean Theorem, a^2 + b^2 = c^2 relates all three sides of this triangle to each other. Let be the height from the top of the ladder to the ground. Let be the distance from the bottom of the ladder to the building. Since 10 is the hypotenuse, we have the following equation.

x^2 + y^2 = 10^2

Simplifying the right side gives us

x^2 + y^2 = 100

Since and are variables, we will wait to plug values into them until after we take the derivative.

The question asks how fast the man standing on the top of the ladder is falling whenthe ladder's base is 6ft from the building and is sliding away at 2 ft/sec. These two values, x=6 and $\frac{\mathrm{d} x}{\mathrm{d} t}=2$ , only happen at a single instant in time. So we will find the derivative of the equation at this point in time.

Using implicit differentiation to find the derivative with respect to time, we get

$2x \frac{\mathrm{d} x}{\mathrm{d} t} + 2y\frac{\mathrm{d} y}{\mathrm{d} t} = 0$

We only care about the instant that x=6 and $\frac{\mathrm{d} x}{\mathrm{d} t}=2$ . We need to find the rate that the top of the ladder, and thus the man, is falling. So we want to solve for $\frac{\mathrm{d} y}{\mathrm{d} t}$ . However, we will need to know what is at this instant in order to find an answer. Fortunately, the Pythagorean Theorem applies at all points in time, so we can use it for this particular instant to find .

6^2 + y^2 = 100

36 + y^2 = 100

y^2 = 64

$y = \pm 8$

Since we are dealing with physical distances, we will only use the positive 8.

Plugging all the information into our derivative equation gives us

$2x \frac{\mathrm{d} x}{\mathrm{d} t} + 2y\frac{\mathrm{d} y}{\mathrm{d} t} = 0$

$2(6)(2) + 2(8)\frac{\mathrm{d} y}{\mathrm{d} t}=0$

$24+ 16\frac{\mathrm{d} y}{\mathrm{d} t}=0$

$16\frac{\mathrm{d} y}{\mathrm{d} t}= -24$

$\frac{\mathrm{d} y}{\mathrm{d} t}= -3/2$

The negative makes sense, because the man is falling down, so the height is getting smaller. Thus our answer is $-3/2 \text{ feet/sec}$

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Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems

The velocity of a car is given by the equation:

$v(t) = 25e^{-t} + 15t$ , where is the time in hours.

If the car starts out at a distance of 3 miles from its home, how far will it be after 4 hours?

Possible Answers:

$98 + \frac{25}{e^{4}}$

$148 + \frac{25}{e^{4}}$

$148 -\frac{25}{e^{4}}$

$145 - \frac{25}{e^{4}}$

$95 + \frac{25}{e^{4}}$

Correct answer:

$148 -\frac{25}{e^{4}}$

Explanation:

To find the total distance traveled, the velocity function has to be integrated from t=0 to t=4 hours:

$distance = \int_{t=0}^{t=4} v(t) dt = \int_{t=0}^{t=4} (25e^{-t} + 15t) dt$

$distance = -25e^{-4} + \frac{15(4)^{2}}{2} - (-25e^{0} + \frac{15(0)^{2}}{2})$

$distance = \frac{-25}{e^{4}} + 120 - (-25 + 0) = \frac{-25}{e^{4}} + 145$

Finally, the question is asking how far the car will be from home. It was 3 miles from home when t=0 , so at t=4 , it will be:

$3 + \frac{-25}{e^{4}} + 145 = 148 -\frac{25}{e^{4}}$ miles from home.

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Example Question #211 : Ap Calculus Ab

A projectile is shot up from a platform $5\ m$ above the ground with a velocity of $150\ m/s$ . Assume that the only force acting on the projectile is gravity that produces a downward acceleration of $9.8\ m/s^{2}$ . Find the height above ground as a f(t) .

Possible Answers:

$y=-4.9t^2+150t+5$

$y=-9.8t^2-150t-5$

$y=-4.9t^2-150t-5$

$y=-9.8t^2+150t+5$

$y=-4.9t^2+150t$

Correct answer:

$y=-4.9t^2+150t+5$

Explanation:

$a=\frac{ds^2}{dt^2}=-9.8$ with initial condition $s(0)=5$

Separate the equation

$ds^2=-9.8dt^2$

Integrate twice

$s=-4.9t^2+150t+c$

Use initial conditions to solve for c

$s=-4.9t^2+150t+5$

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Example Question #212 : Ap Calculus Ab

A gun sends a bullet straight up with a launch velocity of 220 ft/s. It reaches a height of $s=220t-16t^2$ after seconds. How high does the bullet go?

Possible Answers:

$567.25\ ft$

$675.25\ ft$

$657.25\ ft$

$756.25\ ft$

$765.25\ ft$

Correct answer:

$756.25\ ft$

Explanation:

The bullet reaches its max height when its velocity = 0. We can solve for its velocity by taking the derivative of the position function.

$v=220-32t$

v=0 when $t=\frac{55}{8}$

We can then plug that into the position function to solve for a height of 756.25 ft.

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Example Question #213 : Ap Calculus Ab

A gun sends a bullet straight up with a launch velocity of 220 ft/s. It reaches a height of $s=220t-16t^2$ after seconds. When (in seconds) does the bullet hit the ground?

Possible Answers:

6.875

13.75

15.325

10.125

Correct answer:

13.75

Explanation:

The bullet will hit the ground when its position is 0. Using the position equation above we solve for that gives s=0 .

There are two values: t=0 and t=13.75 .

t=0 is when the bullet is initially shot, and t=13.75 when the bullet hits the ground.

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Example Question #1 : How To Find Position

Given the velocity function $v(t)=t^3-t+7$ , find the position at time $x=2$ with initial position $x(0)=2$ .

Possible Answers:

Correct answer:

Explanation:

First, we find the position function $x(t)=\int (t^3-t+7) dt=\frac{t^4}{4}-\frac{t^2}{2}+7t+constant$

$x(0)=2\Rightarrow constant=2\Rightarrow x(t)=\frac{t^4}{4}-\frac{t^2}{2}+7t+2$

Therefore, $x(2)=\frac{2^4}{4}-\frac{2^2}{2}+7*2+2=18$ .

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AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems

Example Question #1 : Modeling Rates Of Change, Including Related Rates Problems

Example Question #2 : Applications Of Derivatives

Example Question #1 : Applications Of Derivatives

Example Question #2 : Applications Of Derivatives

Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems

Example Question #211 : Ap Calculus Ab

Example Question #212 : Ap Calculus Ab

Example Question #213 : Ap Calculus Ab

Example Question #1 : How To Find Position

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AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems­

Example Question #1 : Modeling Rates Of Change, Including Related Rates Problems­

Example Question #2 : Applications Of Derivatives

Example Question #1 : Applications Of Derivatives

Example Question #2 : Applications Of Derivatives

Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems­

Example Question #211 : Ap Calculus Ab

Example Question #212 : Ap Calculus Ab

Example Question #213 : Ap Calculus Ab

Example Question #1 : How To Find Position

All AP Calculus AB Resources

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Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems

Example Question #1 : Modeling Rates Of Change, Including Related Rates Problems

Example Question #4 : Modeling Rates Of Change, Including Related Rates Problems