Chapter 1. calc_tutorial_14_4_023

1.1 Problem Statement

{3,5,7,9}
{2,4,6,8}
{.01,.02,.03,.04}
{.01,.02,.03,.04}
$x+$u
$y+$v
round($a,0)
round($b,0)
round($a-$c,2)
round($b-$d,2)
pow($c,3)*pow($d,2)
3*pow($c,2)*pow($d,2)
2*pow($c,3)*$d
round($g+$h*$e+$i*$f,2)
round(pow($a,3)*pow($b,2),5)
round(abs($j-$ans),5)
round(($k/$j)*100,4)

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

($a)3($b)2

1.2 Step 1

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

Instead of calculating the value f($a, $b) directly, we can use a linear approximation of f(x, y) at ($c, $d) to estimate f($a, $b) since it is very close to f($c, $d). 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 1.1

We will use the above equation to approximate the value of f($a, $b) = ($a)3($b)2 with the nearby point ($c, $d). Therefore the appropriate values we need to input in the above formula are

a = SFgqQUkJGdg=

b = U2GIbglD1oM=

h = 5dWJhLEUU4Y=

k = or6dmYWrEbA=

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Compute the values of f(x, y) = x3y2 and its partial derivatives at ($c, $d).

f($c, $d) = mpgA8Zi4UKU=

fx($c, $d) = Hs0oCnDH0eo=

fy($c, $d) = zDsgB75cNfg=

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

1.4 Step 3

Question Sequence

Question 1.3

Substitute the values from step 2 appropriately into the linear approximation expression to estimate the value of ($a)3($b)2.

($a)3($b)2 = f($c + $e, $d + $f)

≈ f($c, $d) + fx($c, $d)(5dWJhLEUU4Y=) + fy($c, $d)(or6dmYWrEbA=)

= $g + $h(5dWJhLEUU4Y=) + $i(or6dmYWrEbA=)

= qjqZz1N+poo=

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

1.5 Step 4

Question Sequence

Question 1.4

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

($a)3($b)2 = bEz/iYnXpnk=

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

error = |($a)3($b)2 − $ans| = 5VKs1keauzU=

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

Percentage error ≈ error/actual × 100 = Q2IgEMvK+Uo=%

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