Question

Suppose f(x,y)= 4x² + 3y².

a. Calculate the partial derivatives of f.

b. Suppose f(x,y)= 16. Use the implicit function theorem to calculate dy/dx.

c. What is the value of dy/dx if x=1, y=2? 

Answer 1

$f(x,y)=16$

Axis interception points of 16:

Y- intercept:(0,16)

Slope of 16: m=0.

Range: 16

(a)

$f(x,y)=4x^2+3y^2$

$\frac{\delta F}{\delta x}$ is a partial derivative of F w.r.t. x.

$\frac{\delta}{\delta x}(4x^2+3y^2)=8x+0=8x$

Similarly,

$\frac{\delta}{\delta y}(4x^2+3y^2)=0+6y=6y$

(b)

At x=1 and y=2:

$F=4(1)^2+3(2)^2=16.$

Total differential for F:

$=\frac{\delta F}{\delta x}.dx+\frac{\delta F}{\delta y}.dy$

$\delta F=8xdx+6ydy$

When $\delta F=0$

$\implies =8xdx-6ydy=0$

$8xdx=6ydy$

$\frac{dy}{dx}=\frac{8x}{6y}=\frac{4x}{3y}$

(c)

Value of $\frac{dy}{dx}$ when x=1 and y=2:

$=\frac{4(1)}{3(2)}=\frac{4}{6}=\frac{2}{3}$