STEM-Master
Calculus IIIChapter 2: Partial DerivativesSection 2-3

Interpretations of Partial Derivatives — Question 7

PDF ↗

Question 7

Adjacent contours differ by 66, with labels increasing in both positive coordinate directions. Along +x+x they are 22 units apart; along +y+y they are 33 units apart.

Tasks

  1. Estimate both partial derivatives.

  2. Relate spacing to magnitude.

  3. Explain how decreasing labels change signs.

Original worksheet page 1: question and worked solution for 2-3-007
Show solutionHide solution

Question 7 – Solution

Strategy. Divide contour-value change by coordinate distance.

Step 1: Estimates fx≈6/2=3\boxed{f_x\approx 6/2=3} and fy≈6/3=2\boxed{f_y\approx 6/3=2}.

Step 2: Spacing Equal output change over less distance gives larger slope, so the xx trace is steeper.

Step 3: Sign If labels decrease in a positive direction, the corresponding partial is negative. Spacing controls magnitude; label order controls sign.

Original worksheet page 2: question and worked solution for 2-3-007

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

← Section questionsChapter contents