Quiz 1
Two vectors are orthogonal if _.
Their dot product is equal to 0.
What is the general equation of the plane?
a(x-x0) + b(y-y0)-c(z-z0) where (a, b, c) is the normal vector.
When would I use a tree diagram in this course?
Tree diagrams are useful when working with the 2nd case of the chain rule, where you are working with multiple functions of more than one variable.
-For example, if you had a function w(x,y,z) there would be 3 separate branches under w. If x, y, and z are all each functions of three separate variables, they would each have 3 branches (connecting to s, t, and r).
-Tree diagrams are helpful in determining how to construct an equation for a specific partial derivative.
How do I find the critical points of a multivariable function?
Find the partial derivatives (fx and fy) of the function and set them equal to 0. Solve the system of equations.
-Remember that if finding absolute extrema, only include the critical points that fall within the bounds!
You find a unit vector by _.
Dividing a vector by its magnitude.
How do I know if two planes are parallel? How do I know if two planes are orthogonal?
Parallel: Their normal vectors are the same/scalar multiples of each other (they are pointing in the same direction).
Orthogonal: If the dot product of their normal vectors equals to 0, then cos(theta)=0, so there is a 90 degree angle between the planes, making them orthogonal.
What is the equation that I can utilize to find the equation of a tangent plane at a specific point P(x0,y0,z0).
z-z0 = fx(x0, y0)(x-x0) + fy(x0, y0)(y-y0)
where fx(x0, y0) and fy(x0, y0) represent partial derivatives.
Recall that a tangent plane is formed by all possible lines which are tangent to a specific curve at a specific point.
How do I find the volume of f(x,y) if it is continuous on R = [a, b] x [c, d].
Take the double integral of this function!
-In other words, you set up two iterated integrals with bounds of x specified by [a,b] and bounds of y specified by [c,d].
-Then, you integrate by working from the inside --> out.
-Recall that if you integrate with respect to y, you treat x as a constant and vice versa!
How do I know if a set of vectors a, b, and c are coplanar?
Their triple product is equal to 0! Recall that the absolute value of the triple product is equivalent to the volume of the parallelepiped. Triple product: a*(bxc)
where * = dot product
How do I graph the projection of a curve on all 3-coordinate planes? ie. r(t) = (t, t^2, t^3)
To find the projection on the xy-plane, set z = 0.
To find the projection of the xz-plane, set y = 0.
To find the projection of the yz-plane, set x = 0.
You should have 3 separate 2D-graphs.
How do I find the direction of the maximum rate of change of a function? How do I find the magnitude of the maximum rate of change of a function?
- The maximum rate of change of a function is equivalent to the magnitude of its gradient, at a particular point which would be given in the problem.
How do I find maximum & minimum values using Lagrange Multipliers if the function f(x) is bounded by one other function, g(x). (Hint: what systems of equation do I use)?
Gradient f(x,y,z) = lambda * Gradient g(x,y,z)
g(x,y,z) = k
Lambda = Lagrange multiplier
1. Solve the above system of equations (4 unknowns, 4 equations. The 4 equations come from the 3 partial derivates, and g(x,y,z)=k).
2. Plug in all solutions (x,y,z) into f(x,y,z) and identify the maximum and minimum values by evaluating which value is largest/smallest.
How do I find the area of a parallelogram formed by two 3-D vectors?
The length of the cross product vector = area of the parallelogram. To find the length of a vector, find the square root of the sum of all components of the vector squared. ie. for resulting vector a = (a1, a2, a3) |a| = sqrt(a1^2 + a2^2 + a3^2)
What is the formula utilized to find the unit-tangent vector of a vector-valued function (r(t)) at a specific value of t?
T(t) = r'(t)/|r'(t)|
- r'(t) indicates a vector of partial derivatives of r(t), evaluated at t
-|r'(t)| indicates the magnitude of r'(t), evaluated at t (Remember that when you find the magnitude of the vector, your result is a scalar!)
What is a directional derivative? What is the relationship between a directional derivate and a gradient?
- A directional derivate is defined as the rate of change of f(x,y) in the direction of a unit vector u = (a, b)
- Remember that it has to be a unit vector!
- The directional derivate = the dot product of the gradient of the function and this unit vector.
What does Fubini's Theorem state?
-In other words, I can integrate with respect to x first, or with respect to y first, and obtain the same answer.
-Keep in mind that sometimes it will be easier to integrate with respect to one variable rather than another first (ie. you can sometimes avoid having to do integration by parts multiple times).
Are a = (2, 1, 5) and b = (6, 3, 2) orthogonal? How do you know without computing?
No, they are not orthogonal. All components are positive, so it is impossible for their dot product to equal to 0.
Write an equation of the line passing through A = (2, -2, 4) and B = (0, 5, -3). Write your solution in parametric form.
1. v = AB = B-A = (0-2, 5+2, -3-4) = (-2, 7, -7)
2. r = r0 + tv
3. r = (0, 5, -3) + (-2, 7,-7)t ß choose either point A or B to plug in for r0
4. x = -2t, y = 5 +7t, z = -3 + -7t
What are level curves? Describe in words what the level curves of the following function would look like: f(x,y) = (x-1)^2 + (y-2)^2.
To find level curves, set f(x,y) = k for different values of k.
For this function, these level curves would appear like circles centered at (1,2). Their radius would be equal to the sqrt(k).
How can I determine if my critical point is a relative min, relative max, or saddle point (describe in detail!)
Use the 2nd derivative test!
D = fxx*fyy - (fxy)^2
--> Remember to evaluate D at the critical points to test them!
1. If D < 0, the point is a saddle point.
2. If D > 0 and fxx(a, b) > 0, then there is a relative minimum at (a,b).
3. 2. If D > 0 and fxx(a, b) < 0, then there is a relative maximum at (a,b).