### All AP Calculus AB Resources

## 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) **

**Possible Answers:**

1,3,4,and 5

1, 2, and 4

1, 3, and 5

All must be true.

1, and 5

**Correct answer:**

1, 2, and 4

**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,

**Possible Answers:**

The function is not continuous at .

None of the above necessarily

The function is not defined at .

The function is not differentiable at .

**Correct answer:**

The function is not differentiable at .

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?

**Possible Answers:**

They are all differentiable and continuous at

**Correct answer:**

They are all differentiable and continuous at

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?

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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.

This gives you your final answer of

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

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

**Possible Answers:**

**Correct answer:**

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)

**Possible Answers:**

**Correct answer:**

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

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