### All Calculus 1 Resources

## Example Questions

### Example Question #61 : Rate Of Change

The area of a circle is increasing at a rate of . If the area of the circle is , what is the rate of increase of the radius?

**Possible Answers:**

**Correct answer:**

The area of circle in terms of its radius is given as:

This can be rewritten to find the radius:

In this problem, the radius can be found to be:

Now, relate rates of change by deriving each side of the area equation with respect to time:

Solve then for :

### Example Question #62 : Rate Of Change

A triangle is growing taller at a rate of . If it has a base of and height of , how fast is the area of the triangle changing?

**Possible Answers:**

**Correct answer:**

The area of a triangle in terms of its base and height is given by the formula:

To find how the rates of change of each term relate, derive each side of the equation with respect to time:

The base isn't widening, so

Therefore:

### Example Question #61 : How To Find Rate Of Change

A rectangle is inexplicably changing in shape. Its length is growing at a rate of and its width is shrinking at a rate of . If the length is and its width is , what is the rate of change of the area?

**Possible Answers:**

**Correct answer:**

The area of a rectangle is given in terms of its length and width as:

.

The rates of change of each parameter with respect to time can be found by deriving each side of the equation with respect to time:

Therefore, with our known values, it's possible to find the inexplicable rate of change of the rectangle. Remember that the width is shrinking, so its rate should be treated as negative:

### Example Question #64 : How To Find Rate Of Change

The sides of a square are shrinking at an increasing rate. If the rate of rate increase is , the rate of increase is and the sides of a length of , what is the rate at which the rate of growth of the area of the square is changing?

**Possible Answers:**

**Correct answer:**

Note for this problem, we're looking for a change in a rate, so this is dealing with second time derivatives:

Relating the area of a square to length:

Leads to

Now, it's important to note that the term cannot be ignored in this second derivation; treat it like another variable!

Plugging in the given values, this gives:

### Example Question #61 : Rate Of Change

A child is breathing into a bubble wand to create a soap bubble. Treating the bubble as an expanding sphere, If the sphere has a volume of and is growing at a rate of , what is the rate of growth of the sphere's radius?

**Possible Answers:**

**Correct answer:**

Begin this problem by solving for the radius. The volume of a sphere is given as

Solving for the radius:

Now time rate of change between quantities can be found by deriving each side with respect to time:

Solving for the radius rate of time then gives:

### Example Question #64 : How To Find Rate Of Change

A triangle is being stretched in a peculiar way. The base is shrinking at a rate of while the height is increasing at a rate of . If the height of the triangle is and the base of the triangle is , what is the rate of change of the triangle's area?

**Possible Answers:**

**Correct answer:**

The area of a triangle is given by the function:

The relationship between the rate of change of each parameter can be found by deriving each side of the equation with respect to time:

### Example Question #63 : Rate Of Change

The radius of the base of a cone is increasing at a rate of . If the current radius is and its height is , what is the rate of growth of its volume?

**Possible Answers:**

**Correct answer:**

The volume of a cone is given by the formula:

To relate rate of changes over time, derive each side of the equation with respect to time:

Since height isn't changing, , which leaves:

### Example Question #68 : How To Find Rate Of Change

A bar of length is resting against a wall, the ground end from the base of the wall. If the end resting on the ground is moved forward at a rate of , how fast does the end resting on the wall rise? The plane of the ground is perpendicular to the wall.

**Possible Answers:**

**Correct answer:**

The bar resting against the wall forms a right triangle, its length serving as the hypotenuse. This can be used to find the height of the end resting on the wall via the Pythagorean Theorem:

The rates of change can also be related using the Pythagorean Theorem:

Since the length of the bar isn't changing,

Note that the negative value is used because the end on the ground is closing the distance to the wall.

### Example Question #64 : Rate Of Change

A circle is stretching into an ellipse, its horizontal radius expanding at a rate of . If the circle has an area of , what is the rate of growth?

**Possible Answers:**

**Correct answer:**

The area of a circle is given by the formula:

This can be used to find the original radii:

Now, the area of an ellipse in terms of its vertical and horizontal radii is:

To relate rates of change, derive each side with respect to time:

Since the vertical radius does not change, let's designate it as , ;

### Example Question #70 : How To Find Rate Of Change

The shorter leg of a right triangle is growing at a rate of . If the shorter leg has a length of and the hypotenuse has a length of , what is the rate of growth of the angle across from the shorter leg? The hypotenuse is growing, but the longer leg is not.

**Possible Answers:**

**Correct answer:**

The hypotenuse and the shorter leg can be related in terms of the angle across from the leg:

This angle can be found as:

Now, rates of change can be related by deriving each side of the original equation with respect to time:

However, we do not know what is. We can find this by using the Pythagorean theorem and once again deriving:

Since we're told the longer side does not change in length, , leaving

Now, let's return to this previous derivative using our known values and solve for :