# AP Calculus AB : Numerical approximations to definite integrals

## Example Questions

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### Example Question #1 : Integrals

Write the equation of a tangent line to the given function at the point.

y = ln(x2) at (e, 3)

y – 3 = (2/e)(x – e)

y – 3 = ln(e2)(x – e)

y = (2/e)(x – e)

y – 3 = (x – e)

y = (2/e)

y – 3 = (2/e)(x – e)

Explanation:

To solve this, first find the derivative of the function (otherwise known as the slope).

y = ln(x2)

y' = (2x/(x2))

Then, to find the slope in respect to the given points (e, 3), plug in e.

y' = (2e)/(e2)

Simplify.

y'=(2/e)

The question asks to find the tangent line to the function at (e, 3), so use the point-slope formula and the points (e, 3).

y – 3 = (2/e)(x – e)

### Example Question #1 : Integrals

Find the equation of the tangent line at  when

Explanation:

let's go ahead and cancel out the 's. This will simplify things.

this is the slope so let's use the point slope formula.

### Example Question #1 : Numerical Approximations To Definite Integrals

Differentiate

Explanation:

We see the answer is  after we simplify and use the quotient rule.

we could use the quotient rule immediatly but it is easier if we simplify first.

### Example Question #2 : Numerical Approximations To Definite Integrals

Find

Explanation:

When taking limits to infinity, we usually only consider the highest exponents. In this case, the numerator has and the denominator has . Therefore, by cancellation, it becomes as  approaches infinity. So the answer is .

### Example Question #11 : Integrals

Evaluate:

cannot be determined

Explanation:

First, we can write out the first few terms of the sequence , where  ranges from 1 to 3.

Notice that each term , is found by multiplying the previous term by . Therefore, this sequence is a geometric sequence with a common ratio of . We can find the sum of the terms in an infinite geometric sequence, provided that , where  is the common ratio between the terms. Because  in this problem,  is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.

If  then find .

Explanation:

### Example Question #1 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at  on graph

Explanation:

(This is the slope. Now use the point-slope formula)

### Example Question #6 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at (1,1) in

Explanation:

(This is the slope. Now use the point-slope formula.)

If  then

Explanation:

We know that

so,

### Example Question #11 : Integrals

does not exist

2

1

1/4

1/2

2

Explanation:

When we let x = 0 in our original limit, we obtain the 0/0 indeterminate form. Therefore, we can apply L'Hospital's Rule, which requires that we take the derivative of the numerator and denominator separately.

Apply the Chain Rule in the numerator and the Product Rule in the denominator.

If we again substitute x = 0, we still obtain the 0/0 indeterminate form. Thus, we can apply L'Hospital's Rule one more time.

If we now let x = 0, we can evaluate the limit.