AP Calculus BC : Initial Conditions

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #1 : Initial Conditions

The temperature of an oven is increasing at a rate  degrees Fahrenheit per miniute for  minutes. The initial temperature of the oven is  degrees Fahrenheit. 

 

What is the temperture of the oven at ? Round your answer to the nearest tenth. 

Possible Answers:

Correct answer:

Explanation:

Integrating  over an interval  will tell us the total accumulation, or change, in temperature over that interval. Therefore, we will need to evaluate the integral 

to find the change in temperature that occurs during the first five minutes.

A substitution is useful in this case. Let. We should also express the limits of integration in terms of . When , and when  Making these substitutions leads to the integral

.

To evaluate this, you must know the antiderivative of an exponential function.

In general, 

.

Therefore, 

This tells us that the temperature rose by approximately  degrees during the first five minutes. The last step is to add the initial temperature, which tells us that the temperature at  minutes is 

 degrees.

Example Question #1 : Initial Conditions

Find the equation for the velocity of a particle if the acceleration of the particle is given by:

and the velocity at time  of the particle is

Possible Answers:

Correct answer:

Explanation:

In order to find the velocity function, we must integrate the accleration function:

We used the rule

 

to integrate.

Now, we use the initial condition for the velocity function to solve for C. We were told that 

so we plug in zero into the velocity function and solve for C:

C is therefore 30.

Finally, we write out the velocity function, with the integer replacing C:

Example Question #2 : Initial Conditions

Find the work done by gravity exerting an acceleration of   for a   block down  from its original position with no initial velocity. 

Remember that 

, where  is a force measured in  is work measured in , and  and  are initial and final positions respectively. 

Possible Answers:

Correct answer:

Explanation:

The force of gravity is proportional to the mass of the object and acceleration of the object. 

Since the block fell down 5 meters, its final position is  and initial position is .

Example Question #3 : Initial Conditions

Evaluate the following integral and find the specific function which satisfies the given initial conditions:

Possible Answers:

Correct answer:

Explanation:

Evaluate the following integral and find the specific function which satisfies the given initial conditions:

To solve this problem,we need to evaluate the given integral, then solve for our constant of integration.

Let's begin by recalling the following integration rules:

Using these two, we can integrate f(x)

SO, we get:

We are almost there, but we need to find c. To do so, plug in our initial conditions and solve:

So, our answer is:

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