Calculus 3 Flashcards
Domain For Simple Polynomial
D(-inf , inf)
Domain for Fraction With Polynomial in Denominator
Any x-value where the denominator is not equal to zero
Domain of Radical
All values inside radical ≥ 0
Domain of f(x,y) = ln(x+y-1)
y> (-x+1)
Domain of f(x,y) = (√y-x^2) / (1-x^2)
D(x,y) = (-∞, -1] , [-1,1] , [1,∞ )
Domain of f(x,y) = √(x^2 + y^2 -4)
All points where x^2 + y^2 ≥ 4
Do you still apply the chain, product, and quotient rules when doing partial derivatives?
Yes
what does the gradient tell you?
Shows you the direction vector of fastest increase.
Gradient takes a function and spits out a…..
Vector
The formula for finding the directional derivative in a given vector direction?
Dƒ(x,y) = ∇ƒ(x,y) *u
Where u is a unit vector
Linear approximation is given by…
L(x, y, z) = f(x0, y0, z0) + fx(x0, y0, z0)(x − x0) + fy(x0, y0, z0)(y − y0) + fz(x0, y0, z0)(z − z0)
Calculate the directional derivative
f(x,y) = tan^-1 (xy)
v = <1,1> ,
P = (2,5)
- Unit vector = <1÷√2 , 1÷√2 >
- Remember that derivative of inverse tanx = 1 ÷ ((x^2)(y^2) +1)
fx(x,y) at (2,5) = 5/101
fy(x,y) at (2,5) = 2/101
Dƒ(x,y) = ∇ƒ(x,y) *u
Answer = 7/101√2
Unit Vector Formula
u = v/ ||v||
Find the gradient vector at the indicated point.
f(x,y) = xy^2 - yx^2
at P(-1,1)
= y^2 - 2yx
∫∫(x,y) dydx from 0 to pi/2 and 0 to pi
for the function f(x,y) = sin2x cos(6y) dydx
0
Integral of cos^2x
Solution
cos^2x = cos2x/2 +1/2
= sin2x/4+x/2 + c
Double integral x/y^2
Where R= (1,4) x (5,6)
1/8
Double integral of x+y dxdy where xy=6 and x+y=7
125/3
double integral x+2y from 2 to 3 and 1 to 5
32
doublenintegral of rsinø dr from 0 to pi and r = 3 to 4
√19
double integral of e^x^2 from 0 to 1 and y to 1
1/2(e-1)
The double integral of ysinx dy dx from 0 to 1 and the from 0 to pi/2
1/2
double integral y^2x dy dx from x^2 to x and then from 0 to 2
-128/15
Plot the vector field
F(x,y) = <x^2 - y^2 - 4, 2xy>
What is the arc length of a space curve?
s = integral (a to b) [ ||r’(t)|| ] dt
What is the formula to find curvature?
K = ||T’(t)|| / ||r’(t)||
What is the process for finding position by integration, from acceleration?
Integrate to get velocity function plus constant. 2. Set t equal to the provided velocity scalar and solve for constant C=C1 + C2 + C3. 3. Integrate again to get position function plus constant. Set t equal to provided position scalar and solve for constant C=C1 + C2 + C3.
What is the formula for unit tangent vector?
T = r’(t) / ||r’(t)||
How do you draw level curves / contour lines?
Use f(x,y) = c and draw the function for changing values of c.
How do you find partial derivatives using the definition of partial derivatives?
To find F_x, Lim x->0 of f(x+dx, y) - f(x,y) / dx. To find F_y, Lim x->0 of f(x+dx,y) - f(x,y) / dy).
What is the formula for the total differential of the independent variables z?
dz = pdz / pdx * dx + pdz / pdy * dy where pd = partial derivative
How do you find the directional derivative of a function, given a point and a vector?
Find a unit vector for the vector 2. Find the gradient for the function 3. Substitute the point’s coordinates in for x,y,z and find gradient dot u (where u is the unit vector)
How do you find the minimum value of the directional derivative of f(x,y)?
|| grad f(x,y) ||
For the second partials test, when does f have a relative maximum at (a,b)?
If d> 0 and f_xx(a,b) < 0
For the second partials test, when does f have a saddle point?
If d < 0, then (a,b,f(a,b)) is a saddle point.
In spherical coordinates, what is x equal to?
p sin phi cos theta
In spherical coordinates, what is y equal to?
p sin phi sin theta
In spherical coordinates what is p^2 equal to?
x^2 + y^2 + z^2
In spherical coordinates what is tan theta equal to?
y/x
What is the integral template and order of integration for cylindrical coordinates?
intintint (Q) f(x,y,z) = intintint r dz dr dtheta
What is the integral template and order of integration for spherical coordinates?
intintint (q) f(x,y,z) = intintint p^2 sin phi dp dphi dtheta
