Chapter 1. calc_tutorial_14_5_025

1.1 Problem Statement

{2,4,6}
{3,5,7}
round($b/(1+pow($a,2)*pow($b,2)),3)
round($a/(1+pow($a,2)*pow($b,2)),3)
round(($a+$b)/(1+pow($a,2)*pow($b,2)),3)

Calculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector or use Duf(P) = (1/||v||)∇fP · v.

f(x, y) = tan−1(xy), v = <1, 1>, P = ($a, $b)

1.2 Step 1

Question Sequence

Question 1.1

Let u = (v/||v||), v ≠ 0. Recall the formula for the directional derivative in the direction of v.

Duf(x, y) = ∇f(x, y) · u

= (1/||v||)∇f(x,y) · v

Calculate the unit vector, u, in the direction of v.

u = (v/||v||)

= <1, 1>/√XvVM00l89Is=

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

In order to calculate the gradient, we need to first calculate the partial derivatives. Find the partial derivatives, fx and fy at the point P = ($a, $b) and round your answers to three decimal places. Since f(x, y) is a composition of functions, both of these derivatives will require the k3Eu6LwI+Y+z2Ic1mMk9A9F5tDhcfycf7vfYrQ== rule.

fx(x, y) = 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

fx($a, $b) = SFgqQUkJGdg=

fy(x, y) = 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

fy($a, $b) = U2GIbglD1oM=

Calculate the gradient at point P, ∇fP.

∇fP = ∇f($a, $b)

= fx($a, $b), fy($a, $b)

= <SFgqQUkJGdg=, U2GIbglD1oM=>

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Find the directional derivative of f(x, y) in the direction of v at the point P. Round your answer to three decimal places.

Duf($a, $b) = ∇f($a, $b) · u

= (1/√2)<$c, $d> · <1, 1>

= (1/√2)5dWJhLEUU4Y=

Correct.
Incorrect.