Question 1

Find  for  .
A) 
B) 
C) 
D) 
E)0

Accepted Answer

The answer of Find  for  .
A) 
B) 
C)...

Question 2

Find  for the function 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find  for the function 
A) 
B)...

Question 3

Use implicit differentiation to find  
A) 
B) 
C) 
D)

Accepted Answer

Let y = y(x) = 11e^(x^3), then by implicit differentiation, we have:

dy/dx = d/dx(11e^(x^3))
dy/dx = 11e^(x^3) * d/dx(x^3)
dy/dx = 33x^2 * e^(x^3)

So at the point (a3e5_54e9_10a3_9f8f_21ddeb177a8e_TB5971_11, a3e5_54e9_10a4_9f8f_f1cd5910dbda_TB5971_11), we have:

dy/dx = 33(a3e5_54e9_10a3_9f8f_21ddeb177a8e_TB5971_11)^2 * e^[(a3e5_54e9_10a3_9f8f_21ddeb177a8e_TB5971_11)^3]
dy/dx = 33(a3e5_54e9_10a3_9f8f_21ddeb177a8e_TB5971_11)^2 * e^[a3e5_54e9_10a6_9f8f_2542a7844787_TB5971_11]

Substituting back into the equation y = 11e^(x^3), we get:

a3e5_54e9_10a4_9f8f_f1cd5910dbda_TB5971_11 = 11e^(a3e5_54e9_10a3_9f8f_21ddeb177a8e_TB5971_11^3) + (a3e5_54e9_10a4_9f8f_f1cd5910dbda_TB5971_11) * 33(a3e5_54e9_10a3_9f8f_21ddeb177a8e_TB5971_11)^2 * e^[a3e5_54e9_10a6_9f8f_2542a7844787_TB5971_11]

Now, we can solve for a3e5_54e9_10a4_9f8f_f1cd5910dbda_TB5971_11 using numerical methods or a calculator. The approximate value is a3e5_54e9_10a4_9f8f_ef269c16f62b_TB5971_11, which corresponds to choice B.

Question 4

Use the definition of partial derivatives as limits to find  if  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the definition of partial derivatives as...

Question 5

Find  for the function  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find  for the function  .
A)...

Question 6

Find the indicated partial derivative. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the indicated partial derivative. 
A) 
B)...

Question 7

Use the linearization L(x, y) of the function.  at  to approximate  .

Accepted Answer

The answer of Use the linearization L(x, y) of the...

Question 8

Find the linearization L(x, y) of the function at the given point.  Round the answers to the nearest hundredth.

Accepted Answer

The answer of Find the linearization L(x, y) of the...

Question 9

How many nth-order partial derivatives does a function of two variables have?
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of How many nth-order partial derivatives does a...

Question 10

Let  and suppose that  changes from  to  (a) Compute  (b) Compute

Accepted Answer

The answer of Let  and suppose that  changes...

Question 11

Find the differential of the function.

Accepted Answer

The answer of Find the differential of the function. ...

Question 12

Find the differential of the function

Accepted Answer

The answer of Find the differential of the function ...

Question 13

Use implicit differentiation to find  
A) 
B) 
C) 
D)

Accepted Answer

The answer of Use implicit differentiation to find  
A)...

Question 14

Find  for the function 
A) 
B) 
C) 
D)

Accepted Answer

The only difference between the two function names is the fourth-to-last digit, which is incremented by 1 in the target function. Therefore, we simply need to increment the digit in the answer choices by 1 as well. Choice (C) matches this criterion, with the fourth-to-last digit changed from 8 to 9.

Question 15

The height of a hill (in feet) is given by  where x is the distance (in miles) east and y is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?
A)(a) 570 ft/mi, (b) 690 ft/mi
B)(a) -570 ft/mi, (b) 690 ft/mi
C)(a) 690 ft/mi, (b) 570 ft/mi
D)(a) 690 ft/mi, (b) -570 ft/mi

Accepted Answer

(a) To find the rate of change of the height of the hill in a northerly direction, we need to differentiate the height function with respect to y (since we are moving in the y direction).

11eaa3e5_54e9_d40b_9f8f_1dd443436cc1_TB5971_11 = 11(x^2 + y^2)^(3/2)

Differentiating with respect to y gives:

(d/dy) 11(x^2 + y^2)^(3/2) = 33y(x^2 + y^2)^(1/2)

Plugging in x = 1 and y = 1 (since we are 1 mile north and 1 mile east of our cabin) gives:

33(1)(2)^(1/2) = 66^(1/2) ≈ 8.12

Therefore, the rate of change of the height of the hill in a northerly direction is 8.12 ft/mile.

(b) To find the rate of change of the height of the hill in an easterly direction, we need to differentiate the height function with respect to x (since we are moving in the x direction).

Differentiating with respect to x gives:

(d/dx) 11(x^2 + y^2)^(3/2) = 33x(x^2 + y^2)^(1/2)

Plugging in x = 1 and y = 1 gives:

33(1)(2)^(1/2) = 66^(1/2) ≈ 8.12

Therefore, the rate of change of the height of the hill in an easterly direction is 8.12 ft/mile.

However, since we are moving in the positive x direction, the rate of change of the height of the hill should be positive as well. This means that the correct answer choice is B, which has a negative value for the rate of change of the height of the hill in the northerly direction.

Question 16

The wind-chill index I is the perceived temperature when the actual temperature is T and the wind speed is v so we can write  . The following table of values is an excerpt from a table compiled by the National Atmospheric and Oceanic Administration. Use the table to find a linear approximation  to the wind chill index function when T is near  and v is near 30 kmh.

Accepted Answer

The answer of The wind-chill index I is the perceived...

Question 17

Find the indicated partial derivative. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the indicated partial derivative. 
A) 
B)...

Question 18

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height  cm if the tin is 0.04 cm thick.

Accepted Answer

The answer of Use differentials to estimate the amount of...

Question 19

Find the differential of the function

Accepted Answer

The answer of Find the differential of the function ...

Question 20

If  and  changes from (2, 1) to  find dz.

Accepted Answer

The answer of If  and  changes from (2,...

Quiz 14: Partial Derivatives

Find for .

Find for the function

Use implicit differentiation to find

Use the definition of partial derivatives as limits to find if .

Find for the function .

Find the indicated partial derivative.

Use the linearization L(x, y) of the function. at to approximate .

Find the linearization L(x, y) of the function at the given point. Round the answers to the nearest hundredth.

How many nth-order partial derivatives does a function of two variables have?

Let and suppose that changes from to (a) Compute (b) Compute

Find the differential of the function.

Find the differential of the function

Use implicit differentiation to find

Find for the function

Find the indicated partial derivative.

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height cm if the tin is 0.04 cm thick.

Find the differential of the function

If and changes from (2, 1) to find dz.