MATH 251 – LECTURE 13
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 12.7
webAssign: 12.7, due 2/22 11:55 p.m.
Friday 2/19:
Midterm 1: Covering chapters 11 and 12.
Next week: 13.1–4
webAssign: 13.1–3, opens 2/22 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–4 p.m.
or by appointment.
What we’ve covered so far
11.1–4
11.5
11.6
11.7
12.1
12.3–4
12.5–6
12.7
Points, vectors, distances, lengths
Dot product, projections
Cross product
Lines: parametric representations and symmetric equations
Planes: the normal equation
Quadratic surfaces, traces, classification, sketches
Vector functions and space curves, tangent vectors
only Arc length
Functions of several variables, level curves, graphs
Partial derivatives
Tangent planes, differentials, increments
The chain rule
Directional derivatives, the gradient.
Optimization, Local min and max,
Second derivative test, Lagrange multipliers
Lagrange multipliers
Exercise 1. Find the extremal values, and the points where those values are obtained, of the function f (x, y) =
ex(y 2 − 2x) in the compact set x2 + y 2 ≤ 1.
Lagrange multipliers
Exercise 1. Find the extremal values, and the points where those values are obtained, of the function f (x, y) =
ex(y 2 − 2x) in the compact set x2 + y 2 ≤ 1.
Quadratic surfaces - in space
Standard forms:
Ax2 + By 2 + Cz 2 = J
and
Ax2 + By 2 + Cz = 0
Ellipsoid
One-sheeted hyperboloid
Two-sheeted hyperboloid
Cone
Quadratic surfaces - in space
Standard forms:
Ax2 + By 2 + Cz 2 = J
Elliptic paraboloid
and
Ax2 + By 2 + Cz = 0
Hyperbolic paraboloid
Quadratic surfaces - in space
Exercise 2. Find the quadratic equations describing the level curves of the function
f (x, y, z) = log(x2 − 4y 2 + 2z),
and classify these level curves.
Space curves
A space curves is defined by a vector function
r(t) = hf (t), g(t), h(t)i.
Exercise 3. Find a tangent vector to the space curve with function r(t) = hcos(t), sin(t), e−ti at the point given
by t = π.
Space curves
D q
E
3 2
3
Exercise 4. Compute the arc length of r(t) = t , 2 t , t when 0 ≤ t ≤ 1.
Tangent planes
The normal vector of the tangent plane of the surface given by z = f (x, y) is given by n = h−fx0 , −fy0 , 1i.
Exercise 5. Find the equation of the tangent plane of the surface defined by z = log(xy) + 4x3y at the point
(1, 1, 4).
Differentials - increments
Let z = f (x, y), then
∆z = f (x + ∆x, y + ∆y) − f (x, y)
Exercise 6. Find an approximation of
√
99 −
√
and
dz = fx0 dx + fy0 dy.
65 which is better than
√
100 −
√
64 = 10 − 8 = 2.
Directional derivatives
The directional derivative of f in the direction of a vector u (such that |u| = 1) is given by
Du(f ) = ∇f • u.
Exercise 7. Compute the directional derivative of f (x, y) = x2 + y 4 in the direction of v = h2, 2i at the point
P = P (1, 0).