3-D Coordinate System: (12.1)
P : (x1, y1, z1) and Q : (x2, y2, z2), points in 3-space:
(A) Distance Formula: d(P, Q) = (x2 x1)2 + (y2 y1)2 + (z2 z1)2 (B) Midpoint Formula: The midpoint R of the line segment joining P and Q is the point R: x1 + x2 y1 + y2
Math 2433 Week 7 Popper007 1. Have you signed up for a time for your midterm yet? a. YES b. NO (then go do it NOW and come back and choose A)
2. Calculate the partial derivatives.
a) b) c) d) e)
15.1 - DIFFERENTIABILITY AND GRADIENT We say that f is diffe
Math 2433 Week 8 Popper008
15.4 - The Gradient as a Normal; Tangent Lines and Tangent Planes Suppose that f (x, y) is a nonconstant function that is continuously differentiable. That means f is differentiable and its gradient f is continuous. We saw last
Math 2433 Week 9 Popper 009 1. Find the point(s) on the surface at which the tangent plane is horizontal.
a) b) c) d) e) 2. What is today? a) Tuesday
15.7 - Maxima and Minima with Side Conditions Examples: 1. Maximize xy on the el
Math 2433 - Week 10 Notes Popper010 1. Compute the lower Riemann sum for the given function over the interval with respect to the partition
a) b) c) d) e) 2. Sketch the region bounded by the graphs of x + y 2 + 2 = 0and x + y = -4 and find its area.
Math 2433 Notes week 11 (16.4-16.6) Popper011 1. What problem did Leonardo da Vinci, Winston Churchill, Albert Einstein, Thomas Edison and General George Patton have in common? A. All were dyslexic. B. All had false teeth C. They were all blind 2. Which o
Math 2433 Notes week 12 (16.7-16.9) Popper012 1. What day is this? A. Monday
2. Which of the following will represent the volume of the solid bounded above by the plane z = y + 4, below by the xy-plane, and on the sides
Math 2433 Week 13 Notes Popper013 1. "Shambala (or Shambhala)", a mythical Buddhist city said to be located somewhere in the Himalayas, is the subject of a song recorded by which of the following bands? A. The Doors B. Credence Clearwater Revival C. Three
Math 2433 sn 15_1 Review (Dec 4th ) Sign up for final exam time! Get there early no stand-bys. In fact, its a good idea to find out who else has the same exam time and get to CASA about an hour early to study together. Must make 60% on final to PASS If yo
Math 2433 Week 6 Popper006 1. Differentiate
a) b) c) d) e)
14.4 Partial Derivatives The partial derivative of f with respect to x is the function fx obtained by differentiating f with respect to x, treating y as a constant. The partial derivative of f wit
Math 2433 Week 5 Popper005 1. What is today? A. Monday B. Tuesday C. Wednesday D. Thursday E. Friday
2. Express the curvature in terms of t.
14.1 Examples of real-valued functions of two and three variables (read this section!) Finding domain
Math 2433 Week 4 Popper 004 1. What class is this? a. Math 1325 b. Math 1330 c. Math 1431 d. Math 2433
Let (t ) = 3t 2 i +
ln t j e 3t k , find t
(e (e (e (e
1) i+ 1 j+ 1 ( e 3e e 3 ) k 2 3 1) i+ 1 j+ 1 ( e 3e + e3 ) k 2 3 1) i+ 1 ( e
Vector Functions; Calculus: 13.2)
Given functions f1(t), f2(t), f3(t) dened on some t-interval I .
f (t) = (f1(t), f2(t), f3(t)
= f 1 (t ) i + f 2 (t ) j + f 3 (t ) k
is called vector-valued function or vector function.
Given a function of several variables: z = f (x, y ) or w = F (x, y, z )
The gradient of f is the vector: f f i+ j = fx i + fy j f= x y The gradient of F is the vector: F F F i+ j+ k F= x y z = Fx i + Fy j + Fz k
I. Directional Derivatives
f (x, y ) dxdy
= m,n lim
mij xi yj
i=1 j =1 m n
= m,n lim
Mij xi yj
i=1 j =1
where mij = min f (x, y ) and Mij = max f (x, y ) on Rij
= m,n lim
f (xi, yj )xi yj
i=1 j =1
(x1, yj ) an arbitrary point in Rij
Math 2433 Online Midterm Review
Problem 1. (a) Given the points A(1, 2, 1), B (3, 2, 2), C (2, 1, 5) (i) Prove that A, B and C are the vertices of a right triangle. (ii) Determine the length of the hypotenuse of the triangle. (iii) Find an equation for th
Math 2433 Notes Week 1
Welcome! We will start at 9:30. We will start off by going over some of the information you will need for the class.
Popper001 1. What is today? a. Monday b. Tuesday c. Wednesday d. Thursday e. Friday
12.1 Cartesian Space Coordinate
Math 2433 Notes Week 2 Welcome to week 2 session 1! Reminder Quiz 1 due Friday 9/4 at midnight. You have 20 attempts at each quiz. There will be amnesty for midterm and final (thats when I re-open all quizzes a second time) ONLY if all students have attem
Math 2433 Notes Week 3 Popper003 1. Find a set of scalar parametric equations for the line that passes through P(1, 3, -4) and is perpendicular to the xz-plane. a) b) c) d) e) f) None of the above. 2. Find an equation in x, y, z for the plane that passes