Calculus 3

Question 1

Domain For Simple Polynomial

Answer 1

D(-inf , inf)

Question 2

Domain for Fraction With Polynomial in Denominator

Answer 2

Any x-value where the denominator is not equal to zero

Question 3

Domain of Radical

Answer 3

All values inside radical ≥ 0

Question 4

Domain of f(x,y) = ln(x+y-1)

Answer 4

y> (-x+1)

Question 5

Domain of f(x,y) = (√y-x^2) / (1-x^2)

Answer 5

D(x,y) = (-∞, -1] , [-1,1] , [1,∞ )

Question 6

Domain of f(x,y) = √(x^2 + y^2 -4)

Answer 6

All points where x^2 + y^2 ≥ 4

Question 7

Do you still apply the chain, product, and quotient rules when doing partial derivatives?

Answer 7

Yes

Question 8

what does the gradient tell you?

Answer 8

Shows you the direction vector of fastest increase.

Question 9

Gradient takes a function and spits out a…..

Answer 9

Vector

Question 10

The formula for finding the directional derivative in a given vector direction?

Answer 10

Dƒ(x,y) = ∇ƒ(x,y) *u

Where u is a unit vector

Question 11

Linear approximation is given by…

Answer 11

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)

Question 12

Calculate the directional derivative
f(x,y) = tan^-1 (xy)
v = <1,1> ,
P = (2,5)

Answer 12

• 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

Question 13

Unit Vector Formula

Answer 13

u = v/ ||v||

Question 14

Find the gradient vector at the indicated point.
f(x,y) = xy^2 - yx^2
at P(-1,1)

Answer 14

= y^2 - 2yx

Question 15

∫∫(x,y) dydx from 0 to pi/2 and 0 to pi
for the function f(x,y) = sin2x cos(6y) dydx

Answer 15

0

Question 16

Integral of cos^2x

Answer 16

Solution
cos^2x = cos2x/2 +1/2
= sin2x/4+x/2 + c

Question 17

Double integral x/y^2
Where R= (1,4) x (5,6)

Answer 17

1/8

Question 18

Double integral of x+y dxdy where xy=6 and x+y=7

Answer 18

125/3

Question 19

double integral x+2y from 2 to 3 and 1 to 5

Answer 19

32

Question 20

doublenintegral of rsinø dr from 0 to pi and r = 3 to 4

Answer 20

√19

Question 21

double integral of e^x^2 from 0 to 1 and y to 1

Answer 21

1/2(e-1)

Question 22

The double integral of ysinx dy dx from 0 to 1 and the from 0 to pi/2

Answer 22

1/2

Question 23

double integral y^2x dy dx from x^2 to x and then from 0 to 2

Answer 23

-128/15

Question 24

Plot the vector field
F(x,y) = <x^2 - y^2 - 4, 2xy>

Answer 24

Question 25

What is the arc length of a space curve?

Answer 25

s = integral (a to b) [ ||r’(t)|| ] dt

Question 26

What is the formula to find curvature?

Answer 26

K = ||T’(t)|| / ||r’(t)||

Question 27

What is the process for finding position by integration, from acceleration?

Answer 27

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.

Question 28

What is the formula for unit tangent vector?

Answer 28

T = r’(t) / ||r’(t)||

Question 29

How do you draw level curves / contour lines?

Answer 29

Use f(x,y) = c and draw the function for changing values of c.

Question 30

How do you find partial derivatives using the definition of partial derivatives?

Answer 30

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).

Question 31

What is the formula for the total differential of the independent variables z?

Answer 31

dz = pdz / pdx * dx + pdz / pdy * dy where pd = partial derivative

Question 32

How do you find the directional derivative of a function, given a point and a vector?

Answer 32

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)

Question 33

How do you find the minimum value of the directional derivative of f(x,y)?

Answer 33

|| grad f(x,y) ||

Question 34

For the second partials test, when does f have a relative maximum at (a,b)?

Answer 34

If d> 0 and f_xx(a,b) < 0

Question 35

For the second partials test, when does f have a saddle point?

Answer 35

If d < 0, then (a,b,f(a,b)) is a saddle point.

Question 36

In spherical coordinates, what is x equal to?

Answer 36

p sin phi cos theta

Question 37

In spherical coordinates, what is y equal to?

Answer 37

p sin phi sin theta

Question 38

In spherical coordinates what is p^2 equal to?

Answer 38

x^2 + y^2 + z^2

Question 39

In spherical coordinates what is tan theta equal to?

Answer 39

y/x

Question 40

What is the integral template and order of integration for cylindrical coordinates?

Answer 40

intintint (Q) f(x,y,z) = intintint r dz dr dtheta

Question 41

What is the integral template and order of integration for spherical coordinates?

Answer 41

intintint (q) f(x,y,z) = intintint p^2 sin phi dp dphi dtheta