MATH 215 SI

Session 3

Lectures 9 - 11

• 14.1 Functions of Several Variables
• 14.3 Partial Derivatives
• 14.4 Tangent Planes and Linear Approximations

14.1 Functions of Several Variables

A function \(f\) of two variables is a rule that assigns to each ordered pair \((x,y)\) in a set \(D\) a unique real number \(f(x,y)\)

• The set \(D\) is the domain of \(f\), and the set of values \(f\) takes on is the range
• The graph of \(f\) is the surface \(\{(x,y,z) \in \mathbb{R}^3 \mid z = f(x,y),\ (x,y) \in D\}\)

14.1 Functions of Several Variables

cont.

Level Curves and Level Surfaces

• The level curves of \(f(x,y)\) are the curves \(f(x,y) = k\) for constants \(k\) in the range of \(f\)
• A contour map is a collection of level curves
• The level surfaces of \(f(x,y,z)\) are the surfaces \(f(x,y,z) = k\)

Exercises

14.1 Functions of Several Variables

• Sketch a few level curves of \(f(x,y) = y - x^2\)

14.3 Partial Derivatives

The partial derivative of \(f\) with respect to \(x\) at \((a,b)\) is given by \(f_x(a,b) = \lim_{h \to 0} \frac{f(a+h,b) - f(a,b)}{h}\)

• To find \(f_x\), treat \(y\) as a constant and differentiate with respect to \(x\)
• To find \(f_y\), treat \(x\) as a constant and differentiate with respect to \(y\)
• Notation: \(f_x = \frac{\partial f}{\partial x} = \frac{\partial z}{\partial x} = D_x f\)
• \(f_x(a,b)\) is the slope of the tangent line to the trace of the graph of \(f\) in the plane \(y = b\)

14.3 Partial Derivatives

cont.

Higher Derivatives

• Second partial derivatives: \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), \(f_{yy}\)
• Note that \(f_{xy} = (f_x)_y = \frac{\partial^2 f}{\partial y \, \partial x}\)
• Clairaut’s Theorem: if \(f_{xy}\) and \(f_{yx}\) both exist and are continuous in an open region around \((a,b)\), then \(f_{xy}(a,b) = f_{yx}(a,b)\)

Implicit Differentiation

• If \(z\) is defined implicitly by an equation in \(x\), \(y\), \(z\), differentiate both sides with respect to \(x\) (treating \(y\) as constant and \(z\) as a function of \(x\) and \(y\)), then solve for \(\frac{\partial z}{\partial x}\)

Exercises

14.3 Partial Derivatives

Find or estimate \(f_x(0,0)\) and \(f_y(0,0)\) for each of the following functions

• \(f(x,y) = y^2\cos(1 + x - y^2 x)\)
• \(g\) described by the table below

Exercises

14.3 Partial Derivatives cont.

Suppose \(f(x,y)\) is differentiable with continuous derivatives, and \[f(2,5) = 7 \qquad f_x(2,5) = 3 \qquad f_y(2,5) = -2\]

• Let \(C\) be the curve of intersection of the plane \(x = 2\) and the surface \(z = f(x,y)\)
• Find a parametric equation of the line that lies in the plane \(x = 2\) and is tangent to \(C\) at \((2, 5, 7)\)

14.4 Tangent Planes and Linear Approximations

Suppose \(f\) has continuous partial derivatives. The tangent plane to the surface \(z = f(x,y)\) at the point \((a, b, f(a,b))\) is given by \[ z - f(a,b) = f_x(a,b)(x - a) + f_y(a,b)(y - b) \]

• Compare this to the tangent line \(y - f(a) = f'(a)(x - a)\) from single variable calculus
• The linearization of \(f\) at \((a,b)\) is \(L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)\)
• For \((x,y)\) near \((a,b)\) we have the linear approximation \(f(x,y) \approx L(x,y)\)

14.4 Tangent Planes and Linear Approximations

cont.

Differentiability and Differentials

• If \(f_x\) and \(f_y\) exist near \((a,b)\) and are continuous at \((a,b)\), then \(f\) is differentiable at \((a,b)\)
• Existence of the partial derivatives alone does not guarantee differentiability
• The total differential of \(z = f(x,y)\) is given by \(dz = f_x(x,y)\,dx + f_y(x,y)\,dy\)
• \(dz\) approximates the actual change \(\Delta z\) when \(dx = \Delta x\) and \(dy = \Delta y\) are small

Exercises

14.4 Tangent Planes and Linear Approximations

Suppose \(g(x,y) = x + \ln(5x^2 - 4y^2)\)

• Find an equation for the tangent plane to the surface \(z = g(x,y)\) at the point \((1, 1, 1)\)
• Find the linearization of \(g\) at \((1,1)\), and use it to approximate \(g(1.02, 0.99)\)

Bonus

Which of the following equations does \(z = f(x+t) + g(x-t)\) satisfy for all differentiable single-variable functions \(f(s)\) and \(g(s)\)?

• 1. \(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial t} = 0\)
• 2. \(\frac{\partial z}{\partial x} = \frac{\partial z}{\partial t}\)
• 3. \(\frac{\partial z}{\partial x} + \frac{\partial^2 z}{\partial t^2} = 0\)
• 4. \(\frac{\partial z}{\partial x} = \frac{\partial^2 z}{\partial t^2}\)
• 5. \(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial t^2} = 0\)
• 6. \(\frac{\partial^2 z}{\partial x^2} = \frac{\partial^2 z}{\partial t^2}\)

Fin

This concludes session 3.