# AP Calculus AB : Derivatives

## Example Questions

### Example Question #1 : Relationship Between Differentiability And Continuity

The function  is differentiable at the point . List which of the following statements must be true about :

1)   The limit     exists.

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2)

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3)

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4)

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5)

1,3,4,and 5

1, 2, and 4

1, 3, and 5

All must be true.

1, and 5

1, 2, and 4

Explanation:

1) If a function is differentiable, then by definition of differentiability the limit defined by,

exists. Therefore (1) is required by definition of differentiability. _______________________________________________________________

2) If a function is differentiable at a point then it must also be continuous at that point. (This is not conversely true).

For a function to be continuous at a point  we must have:

Therefore (2) and (4) are required.

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3)

This is not required, the left side of the equation is the definition of a derivative at a point  for a function . The derivative at a point does not have to equal to the function value  at that point, it is equal to the slope  at that point. Therefore 3 does not have to be true.

However, we can note that it is possible for a function and its' derivative to be equal for a given point. Sine and cosine, for instance will intersect periodically. Another example would be the exponential function  which has itself as its' derivative

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4) See 2

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5)

Again, the function does not have to approach the same limit as its' derivative. It is possible for a function to behave in this manner, such as in the case of sine and its' derivative cosine, which will both have the same limit at points where they intersect.

### Example Question #2 : Concept Of The Derivative

When the limit  fails to exist,

The function  is not continuous at .

None of the above necessarily

The function  is not defined at .

The function  is not differentiable at .

The function  is not differentiable at .

Explanation:

By definition of differentiability,  when the limit exists. When  exists, we say the function is 'differentiable at '.

### Example Question #3 : Concept Of The Derivative

Which of the following functions is differentiable at ,  but not continuous there?

They are all differentiable and continuous at

They are all differentiable and continuous at

Explanation:

All of the functions are differentiable at . If you examine the graph of each of the functions, they are all defined at , and do not have a corner, cusp, or a jump there; they are all smooth and connected (Not necessarily everywhere, just at ). Additionally it is not possible to have a function that is differentiable at a point, but not continuous at that same point; differentiablity implies continuity.

### Example Question #4 : Concept Of The Derivative

For which of the following functions does a limit exist at , but not a y-value?

Explanation:

To answer the question, we must find an equation which satisfies two criteria:

(1) it must have limits on either side of  that approach the same value and (2) it must have a hole at .

Each of the possible answers provide situations which demonstrate each combination of (1) and (2). That is to say, some of the equations include both a limit and a y-value at neither, or,in the case of the piecewise function, a y-value and a limit that does not exist.

In the function, , the numerator factors to

while the denominator factors to . As a result, the graph of this

function resembles that for , but with a hole at . Therefore, the limit

at  exists, even though the y-value is undefined at .

### Example Question #1 : Slope Of A Curve At A Point

Find the slope of the curve  at the point with x-coordinate .

Explanation:

To find the slope at a point, we take the derivative of , substitute in , and simplify.

.

.

### Example Question #2 : Slope Of A Curve At A Point

Find the slope of at t=0.

Explanation:

First, use the chain rule to find f'(t).

You should get .

Next, plug in t=0 to get f'(0)=.

### Example Question #1 : Slope Of A Curve At A Point

Find the slope of the curve , at .

Explanation:

Find the derivative of the curve using the power rule.

In mathematical terms, the power rule states,

Therefore the derivative is,

Next, plug in the x-value to find the slope of the curve at , which gives you a final answer of

### Example Question #4 : Slope Of A Curve At A Point

Find the slope of the curve, , at .

Explanation:

Expand the binomial and combine like terms to get,

Next, take the derivative of the polynomial using the power rule, which is in mathematical terms,

Therefore,

Lastly, plug in  for , to get the slope at the point.

### Example Question #5 : Slope Of A Curve At A Point

Find the slope at the point (1,2) of the function

Explanation:

Derivatives are slope finders. Therefore to find the slope at the given point, we need to find the derivative of the function using power rule. Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent.

So, we get

From there we plug in our x value.

### Example Question #6 : Slope Of A Curve At A Point

Find m in  from the equation, given the point (2,0)

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function.

To find the derivative we need to use product rule. Product rule states that we take the derivative of the first function and multiply it by the derivative of the second function and then add that with the derivative of the second function multiplied by the given first function. To find the derivative of each seperate function we need to use power rule.

Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent

Use power rule and we get :

From here, to find the slope at the given point we plug in "2" for x.

This comes out to equal