AP Calculus AB › Calculus
If , what is
?
The given function consists of a function, , inside another function,
, such that
.
Thus we can use the Chain Rule to find .
The Chain Rule says that if
,
then
.
Recall (or look up) that the derivative of sine is cosine, so , and use the Power Rule to get
.
Combining the three functions ,
, and
, we have
.
Note: The Power Rule says that for a function
,
.
The radius of a sphere and the sides of a cube begin to grow at the same rate. What must the ratio of the radius' length to the sides' length be such that the rate of growth of the two shapes' volumes is equal?
Begin by writing the volume equations for a sphere and a cube:
The rate of change of these volumes can be found by taking the derivatives of these equation with respect to time:
We're given the relationship between the rate of expansion of the pertinent lengths for these shapes, namely .
To find the necessary ratios of lengths for the volumes to have equal rates of expansion, set the volume rate expressions equal to each other:
Find the position function if the velocity function is
and .
In order to find the position function from the velocity function we need to find the anti-derivative of the velocity function
When taking the integral we use the inverse power rule which states
Applying this rule we get
To find the value of the constant we use the initial condition
which yields
Therefore
What is the area below and above the
-axis?
To find the area below a curve, you must find the definite integral of the function. In this case the limits of integration are where the original function intercepts the -axis at
and
. So you must find
which is
evaluated from
to
. This gives an answer of
The rate of growth of the population of wild foxes in Britain is proportional to the population. The population increased by 113 percent between 2011 and 2015. What is the constant of proportionality in years-1?
We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:
Where is an initial population value, and
is the constant of proportionality.
Since the population increased by 113 percent between 2011 and 2015, we can solve for this constant of proportionality:
The spherical bottom of a snowman is melting such that its volume is decreasing at a rate of .
What is the rate of change of the radius, to the nearest whole number, when the radius is ?
First, find the rate of increase in the radius :
Next, take the derivative of the area of a circle () by using the Power Rule (
):
Lastly, plug in the givens and simplify:
A spherical balloon is being filled with air. What is the volume of the sphere at the instance the rate of growth of the volume is an eighth of the rate of growth of the surface area?
Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:
The rates of change can be found by taking the derivative of each side of the equation with respect to time:
The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is an eighth of the rate of growth of the surface area, let's solve for a radius that satisfies it.
Then to find the volume:
Utilize the method of midpoint Riemann sums to approximate using three midpoints.
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles each with a base of length
and variable heights
, which depend on the function value at a given point
.
We're asked to approximate
So the interval is , the subintervals have length
, and since we are using the midpoints of each interval, the x-values are
A cube is growing in size. What is the ratio of the rate of growth of the cube's volume to the rate of growth of the area of a single face when its sides have length 93?
Begin by writing the equations for a cube's dimensions. Namely its volume and the area of a face in terms of the length of its sides:
The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:
Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and surface area:
Using the method of midpoint Reimann sums, approximate the integral using four midpoints.
A Reimann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles each with a base of length
and variable heights
, which depend on the function value at a given point
.
We're approximating
So the interval is , the subintervals have length
, and since we are using the midpoints of each interval, the x-values are