calc_tutorial_14_4_023

 
Problem Statement

{3,5,7,9}
{2,4,6,8}
{.01,.02,.03,.04}
{.01,.02,.03,.04}
9+.02
8+.03
round(9.02,0)
round(8.03,0)
round(9.02-9,2)
round(8.03-8,2)
pow(9,3)*pow(8,2)
3*pow(9,2)*pow(8,2)
2*pow(9,3)*8
round(46656+15552*0.02+11664*0.03,2)
round(pow(9.02,3)*pow(8.03,2),5)
round(abs(47320.65018-47316.96),5)
round((3.69018/47320.65018)*100,4)

Use linear approximation to estimate the value. Compare with the value given by a calculator. State the percentage error.

(9.02)3(8.03)2

 
Step 1

The number (9.02)3(8.03)2 is a value of the function f(x, y) = x3y2.

Instead of calculating the value f(9.02, 8.03) directly, we can use a linear approximation of f(x, y) at (9, 8) to estimate f(9.02, 8.03) since it is very close to f(9, 8). There are several ways to write the linear approximation to f(x, y) at a point (a, b) but the most useful method here is the following since we wish to approximate the value of a nearby point.

f(a + h, b + k) ≈ f(a, b) + fx(a, b)h + fy(a, b)k

Question Sequence

Question 1

We will use the above equation to approximate the value of f(9.02, 8.03) = (9.02)3(8.03)2 with the nearby point (9, 8). Therefore the appropriate values we need to input in the above formula are

a =

b =

h =

k =

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

Compute the values of f(x, y) = x3y2 and its partial derivatives at (9, 8).

f(9, 8) =

fx(9, 8) =

fy(9, 8) =

Incorrect. Do you see where you went wrong?
Excellent work.

 
Step 3

Question Sequence

Question 3

Substitute the values from step 2 appropriately into the linear approximation expression to estimate the value of (9.02)3(8.03)2.

(9.02)3(8.03)2 = f(9 + 0.02, 8 + 0.03)

≈ f(9, 8) + fx(9, 8)() + fy(9, 8)()

= 46656 + 15552() + 11664()

=

Incorrect. Do you see where you went wrong?
Excellent work.

 
Step 4

Question Sequence

Question 4

Use a calculator to find the value of (9.02)3(8.03)2 to five decimal places.

(9.02)3(8.03)2 =

Find the error in the estimate found using linear approximation. (Round your answer to five decimal places.)

error = |(9.02)3(8.03)2 − 47316.96| =

State the percentage error. (Round your answer to four decimal places.)

Percentage error ≈ error/actual × 100 = %

Incorrect. Do you see where you went wrong?
Excellent work.