Exercise 7, Page 720

Find the total differential dz for z = 5x − 7y



\(dz = f_x(x, y)dx + f_y(x, y)dy\)

\(f_x(x, y)dx = 5\)

\(f_y(x, y)dy = -7\)

Therefore:

\[ \begin{split} dz &= f_x(x, y)dx + f_y(x, y)dy \\ &= 5 + (-7) \\ &= \boxed{ -2 } \end{split} \]

Exercise 9, Page 728

In Exercises 7 – 12, funcions z = f(x, y), x = g(t) and y = h(t) are given.

  1. Use the Multivariable Chain Rule to compute \(\frac{dz}{dt}\).

  2. Evaluate \(\frac{dz}{dt}\) at the indicated t-value.

  1. z = 5x + 2y, x = 2 cos t + 1, y = sin t - 3; t = π/4

Compute \(\frac{dz}{dt}\) first:

\(z = 5x + 2y,\)

\(x = 2 cos\ t + 1\),

\(y = sin\ t - 3\),

\(f_x(x,y) = 5\),

\(f_y(x,y) = 2\)

\(\frac{dx}{dt} = -2 sin\ t\)

\(\frac{dy}{dt} = cos\ t\)

Hence:

\[ \begin{split} \frac{dz}{dt} &= \frac{dz}{dx}\frac{dx}{dt} + \frac{dz}{dy}\frac{dy}{dt} \\ &= 5(-2 sin\ t) + 2(cos\ t) \\ &= \boxed{ -10 sin\ t + 2 cos\ t } \end{split} \]

For \(t =\frac{\pi}{4}\), we evaluate \(\frac{dz}{dt}\) using R like so:

t <- pi/4
dzdt <- -10*sin(t) + 2*cos(t)
dzdt
## [1] -5.656854