AP Calculus BC : AP Calculus BC

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #1 : Parametric Form

Find dy/dx at the point corresponding to the given value of the parameter without eliminating the parameter:



Possible Answers:

Correct answer:

Explanation:

The formula for dy/dx for parametric equations is given as:

From the problem statement:

If we plug these into the above equation we end up with:

If we plug in our given value for t, we end up with:

This is one of the answer choices.

Example Question #7 : Parametric, Polar, And Vector Functions

Draw the graph of  from .

Possible Answers:

R_sinx

Faker_cosx

R_sinx_1

R_sin2x

R_cosx

Correct answer:

R_sinx

Explanation:

Between  and , the radius approaches  from .

From  to  the radius goes from  to .

Between  and , the curve is redrawn in the opposite quadrant, the first quadrant as the radius approaches .

From  and , the curve is redrawn in the second quadrant as the radius approaches  from .   

Example Question #8 : Parametric, Polar, And Vector Functions

Draw the graph of  where .

Possible Answers:

R_sinx_1

R_sinx

R_sin2x

R_cos2x

Faker_cosx

Correct answer:

R_sin2x

Explanation:

Because this function has a period of , the amplitude of the graph   appear at a reference angle of  (angles halfway between the angles of the axes).  

Between  and  the radius approaches 1 from 0.

Between  and , the radius approaches 0 from 1.

From  to  the radius approaches -1 from 0 and is drawn in the opposite quadrant, the fourth quadrant because it has a negative radius.

Between  and , the radius approaches 0 from -1, and is also drawn in the fourth quadrant.

From  and , the radius approaches 1 from 0. Between  and , the radius approaches 0 from 1.

Then between  and  the radius approaches -1 from 0. Because it is a negative radius, it is drawn in the opposite quadrant, the second quadrant. Likewise, as the radius approaches 0 from -1. Between  and , the curve is drawn in the second quadrant.                  

Example Question #781 : Calculus Ii

Graph  where .

Possible Answers:

R_cosx

R2_cos2x

R_sin2x

R_cos2x

R2_sin2x

Correct answer:

R2_cos2x

Explanation:

Taking the graph of , we only want the areas in the positive first quadrant because the radius is squared and cannot be negative.

This leaves us with the areas from  to  to , and  to .

Then, when we take the square root of the radius, we get both a positive and negative answer with a maximum and minimum radius of .

To draw the graph, the radius is 1 at  and traces to 0 at . As well, the negative part of the radius starts at -1 and traces to zero in the opposite quadrant, the third quadrant.

From  to , the curves are traced from 0 to 1 and 0 to -1 in the fourth quadrant. Following this pattern, the graph is redrawn again from the areas included in  to .    

Example Question #201 : Ap Calculus Bc

Draw the curve of  from .

Possible Answers:

R_sinx

R_sinx_1

R2_sin2x

R2_cos2x

R_sin2x

Correct answer:

R2_sin2x

Explanation:

Taking the graph of , we only want the areas in the positive first quadrant because the radius is squared and cannot be negative.

This leaves us with the areas from  to  and  to 

Then, when we take the square root of the radius, we get both a positive and negative answer with a maximum and minimum radius of .

To draw the graph, the radius is 0 at  and traces to 1 at . As well, the negative part of the radius starts at 0 and traces to-1 in the opposite quadrant, the third quadrant.

From  to , the curves are traced from 1 to 0 and -1 to 0 in the third quadrant.

Following this pattern, the graph is redrawn again from the areas included in  to .    

Example Question #161 : Parametric, Polar, And Vector

Rewrite in polar form:

Possible Answers:

Correct answer:

Explanation:

Example Question #1 : Polar Form

What is the following coordinate in polar form?

Provide the angle in degrees.

Possible Answers:

Correct answer:

Explanation:

To calculate the polar coordinate, use

However, keep track of the angle here. 68 degree is the mathematical equivalent of the expression, but we know the point (-2,-5) is in the 3rd quadrant, so we have to add 180 to it to get 248.

Some calculators might already have provided you with the correct answer.

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Example Question #201 : Ap Calculus Bc

What is the equation  in polar form?

Possible Answers:

Correct answer:

Explanation:

We can convert from rectangular form to polar form by using the following identities:  and . Given , then .

. Dividing both sides by ,

 

Example Question #3 : Polar Form

What is the equation  in polar form?

Possible Answers:

None of the above

Correct answer:

Explanation:

We can convert from rectangular form to polar form by using the following identities:  and . Given , then . Multiplying both sides by ,

Example Question #4 : Polar Form

Convert the following function into polar form:

Possible Answers:

Correct answer:

Explanation:

The following formulas were used to convert the function from polar to Cartestian coordinates:

Note that the last formula is a manipulation of a trignometric identity.

Simply replace these with x and y in the original function.

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