Test 2: 12.5 to 13.8

Question 1

What is the arc length of a space curve?

Answer 1

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

Question 2

What is the formula to find curvature?

Answer 2

K = ||T’(t)|| / ||r’(t)||

Question 3

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

Answer 3

1. 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 4

What is the formula for unit tangent vector?

Answer 4

T = r’(t) / ||r’(t)||

Question 5

What is the formula for unit normal vector?

Answer 5

N = T’(t) / ||T’(t)||

Question 6

How do you draw level curves / contour lines?

Answer 6

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

Question 7

How do you find the limit of a function of x and y?

Answer 7

Set x equal to zero and find the limit. Set y equal to zero and find the limit. Set y equal to x and find the limit. If either of the limits don’t match the final one, the limit does not exist.

Question 8

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

Answer 8

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 9

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

Answer 9

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

Question 10

What is the process to find maximum possible error in a measurement?

Answer 10

1. Find the equation for the desired measurement. 2. Take the total differential of of the desired variable representing the desired measurement, using the provided possible error for dx/dy/dz. 3. Divide the total differential by the actual desired measurement, substituting in provided measurement components. Do not use negatives.

Question 11

What is the chain rule for one independent variable?

Answer 11

dw/dt = pdw/pdx * dx/dt + pdw/pdy * dy/dt

Question 12

What is the chain rule for two independent variables?

Answer 12

pdw/pds = pdw/pdx * pdx/pds + pdw/pdy * pdy/pds, and pdw/pdt = pdw/pdx * pdx/pdt + pdw/pdy * pdy/pdt

Question 13

How do you draw a tree?

Answer 13

Start at the top and draw the dependent variable (usually w), draw x and y and z under w, and draw t(and s if necessary) under x,y and z.

Question 14

How do you find dy/dx using implicit differentiation?

Answer 14

dy/dx = -F_x(x,y) / F_y(x,y)

Question 15

How do you find pdz/pdx and pdz/pdy using implicit differentiation?

Answer 15

pdz/pdx = -F_x(x,y,z)/F_z(x,y,z). pdz/pdy = -F_y(x,y,z)/F_z(x,y,z)

Question 16

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

Answer 16

1. 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 17

What is the gradient of a function?

Answer 17

gradf(x,y) = f_x(x,y)i + f_y(x,y)j

Question 18

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

Answer 18

|| grad f(x,y) ||

Question 19

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

Answer 19

• || grad f(x,y) ||

Question 20

What is the equation of a tangent plane?

Answer 20

F_x(x0,y0,z0)(x-x0) + F_y(x0,y0,z0)(y-y0) + F_z(x0,y0,z0)(z-z0) = 0

Question 21

How do you find the equation of a normal line?

Answer 21

1. Find the gradient to the surface 2. Plug in the provided point 3. Use the symmetric line equations x-x0 / v1 = y-y0 / v2 = z-z0 / v3

Question 22

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

Answer 22

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

Question 23

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

Answer 23

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

Question 24

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

Answer 24

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

Question 25

When is the second partials test inconclusive?

Answer 25

When d = 0.

Question 26

What is the equation for the second partials test?

Answer 26

d = f_xx(a,b)f_yy(a,b)-[f_xy(a,b)]^2