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 
What is the temperture of the oven at 
Integrating 
to find the change in temperature that occurs during the first five minutes.
A substitution is useful in this case. Let
To evaluate this, you must know the antiderivative of an exponential function.
In general,
Therefore,
This tells us that the temperature rose by approximately 
Example Question #2 : 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 
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 #3 : Initial Conditions
Find the work done by gravity exerting an acceleration of 
Remember that
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 
Example Question #1 : Application Of Integrals
Evaluate the following integral and find the specific function which satisfies the given initial conditions:
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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