Hyperbolic Functions
Certain combinations of the exponential function occur so often in physical
applications that they are given special names. Specically, half the dierence of
ex and ex is dened as the hyperbolic sine function and half their sum is the
Notes on Series Solutions of Second-Order Linear Ordinary Dierential Equations
We are given a second-order linear dierential equation in y = y(x) for which x0 = 0 is an ordinary
point. We wish to nd solutions in the form of innite series about x0 = 0; i.e
The Chain Rule
Suppose z = f (x, y) is a dierentiable function
of x and y, where x = g(t) and y = h(t) are both differentiable functions of t. Then z is a dierentiable
function of t and
z dx z dy
dz
.
+
=
y dt
x dt
dt
The following graph gives an illustra
Tangent Planes to Surfaces
Let F be a dierentiable function of three variables x, y, and z. For a constant k, the equation
F (x, y, z) = k represents a surface S in space. For
example, the equation x2 + y 2 + z 2 = 9 represents
the sphere with radius 3 an
Simon Fraser University MATH 130 - Geometry for Computer Graphics
Spring term 2014 Instructor: Kirill Pichon Gostaf
January 31, 2014
Midterm #1.A - 50 minutes
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