# Calculus 1 : How to find rate of change

## Example Questions

### Example Question #41 : How To Find Rate Of Change

The seconds hand in the clock shown has a length of five inches. Find the velocity of its tip in the  and  directions at the depicted moment in time.

Explanation:

For this problem, it will be useful to work in radians. The angle of the secondshand, between the positive x-axis and itself, is given as:

Now, to find the x and y velocities, first write out the form of their positions:

Therefore, their respective velocities are given by their respective derivatives:

The secondshand doesn't change in length, so . As for , that can be found knowing how long it takes the secondshand to complete the circuit around the clock:

Note that movement in the clockwise direction, when dealing with angular motion, is treated as negative.

Knowing this, we can find the velocities:

### Example Question #41 : How To Find Rate Of Change

A metal cylinder with a radius of five inches and a height of twenty inches is being heated, causing it to expand. If the radius grows at a rate of   and the height grows at a rate of , what is the rate of expansion of the cylinder's volume?

Explanation:

The volume of a cylinder is given by the formula:

wherein  represents the cylinder's radius, and  its height.

Therefore, to find the rate of expansion, derive each side of the equation with respect to time:

Therefore, for the values given in the problem statement, the rate of expansion of the volume can be found:

### Example Question #41 : Rate Of Change

The sides of a right triangle are increasing in length. If the shorter side, which has a length of , increases at a rate of , and the longer side, which has a length of , increases at a rate of , what is the rate of growth of the hypotenuse?

Explanation:

Since this is a right triangle, the hypotenuse can be related to the sides utilizing the pythagorean theorem:

Although the root of each side may be taken, it would be simpler to take the derivative of each side with respect to time as is:

Therefore, the rate of change of the hypotenuse is:

Using the Pythagorean Theorem, the hypotenuse is:

Thus:

### Example Question #1 : Comparing Relative Magnitudes (Exponential Growth,Logarithmic Growth, Polynomial Growth)

A cylinder of height  and radius  is expanding. The radius increases at a rate of  and its height increases at a rate of . What is the rate of growth of its surface area?

Explanation:

The surface area of a cylinder is given by the formula:

To find the rate of growth over time, take the derivative of each side with respect to time:

Therefore, the rate of growth of surface area is:

### Example Question #42 : Rate Of Change

Water is being poured into a cylindrical glass at a rate of . If the cylinder has a diameter of  and a height of , what is the rate at which the water rises?

Explanation:

The volume equation of a cylinder is

Therefore, the rate of change of each term can be related by taking the derivative of each side of the equation:

Treating the glass as solid, the radius of liquid in it will not change, only the height, so . This simplifies the equation:

Since the radius is half the diameter:

### Example Question #41 : Rate Of Change

The position in meters of a particle after  seconds is modeled by the equation

, where .

At what rate, in meters per second, is the position of the particle changing at  seconds?

Explanation:

The derivative of a function gives us a new function that describes the rate of change of the original function at every point in its domain. Therefore the derivative of f(t) will produce a new function that describes the rate of change in position with respect to time. Using the product rule, the derivative of f(t) is:

Next substitute 3 for to find the rate of change of the particle at exactly 3 seconds.

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### Example Question #1931 : Functions

What is the rate of change of the function

when ?

Explanation:

The find the rate of change of a function at a particular point, find the derivative of the function, then substitute the point into the function. Using the quotient rule, the derivative is:

Next substitute  for x.

Now we need to simplify the right hand side.

### Example Question #45 : Rate Of Change

What is the rate of change of a square's sides if it has an area of  which is growing at a rate of ?

Explanation:

The area of a square in terms of the lengths of its sides is given as:

Therefore the length of each side can be found to be

Rates of change can also be related by taking the time derivative of each side:

The rate of change of the sides, , is what is being solved for:

### Example Question #48 : How To Find Rate Of Change

Find the derivative of the following function through implicit differentiation:

Explanation:

By differentiating both sides:

By solving for :

### Example Question #46 : Rate Of Change

Air flows out of a spherical balloon at a rate of . What is the rate of change of the circumference when the radius of the balloon is  units?