Partial Derivatives

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AP Calculus BC › Partial Derivatives

Questions 1 - 10
1

Explanation

2

Evaluate the limit

Explanation

The limiting situation in this equation would be the denominator. Plug the value that n is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when n=1; so we try to eliminate the denominator by factoring. When the denominator is no longer zero, we may continue to insert the value of n into the remaining equation.

3

Find the total derivative of the following function:

Explanation

The total derivative of a function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

, ,

4

Evaluate the limit:

Explanation

To evaluate the limit, we must factor out a term consisting of the highest power term divided by itself (which equals one, so we aren't changing the original function):

The term we factored goes to one, and the two terms with negative exponents in the denominator go to zero (they are each "fractions" with n in their denominator - the terms go to zero as the denominator goes to infinity), so we are left with .

5

Find of the function

Explanation

To find of the function, you take two consecutive partial derivatives:

6

Evaluate the limit

Explanation

The limiting situation in this equation would be the denominator. Plug the value that n is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when n=1; so we try to eliminate the denominator by factoring. When the denominator is no longer zero, we may continue to insert the value of n into the remaining equation.

7

Explanation

8

Explanation

9

What is the partial derivative of the function

?

Explanation

We can find given by differentiating the function while holding constant, i.e. we treat as a number. So we get

10

Evaluate the following limit:

Explanation

To evaluate the limit, we must first pull out a factor consisting of the highest power term divided by itself (one, essentially):

After the factor we created becomes 1, the negative exponent terms go to zero as x approaches infinity, therefore we are left with .

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