Question

A two-dimensional temperature profile is given as \( T = 2x^2 + 3xy + y^2 \). If \(\hat{\imath}\) and \(\hat{\jmath}\) are the unit vectors along x and y directions, respectively, which one of the following is the directional derivative of T at the location x = 2, y = 2?

• A. \( 8\hat{\imath} + 12\hat{\jmath} \)
• B. \( 14\hat{\imath} + 10\hat{\jmath} \)
• C. \( 8\hat{\imath} - 12\hat{\jmath} \)
• D. \( 14\hat{\imath} - 10\hat{\jmath} \)

Hint:

Compute partial T/partial x = 4x+3y and partial T/partial y = 3x+2y, then substitute x=2, y=2 into both.

Answer 1

The Correct Option is B

Using the limit definition: T(2,2)=24, T(2+h,2)=24+14h+2h^2, giving partial T/partial x = 14. Similarly T(2,2+k)=24+10k+k^2, giving partial T/partial y = 10. This confirms the gradient by direct limit calculation instead of symbolic differentiation.

\[ 14\hat{\imath} + 10\hat{\jmath} \ \Rightarrow \ \text{Option (B)} \]