MATH 53

Sethian

a) 288
b) 144

a) (1/2) e^-1/2 + (3/2) e^-1 + e^-2

b) (1/4) e^-1/8 + e^-3/8 + (3/4) e^-9/8

3

f(x, y) = x + 3x²y²

2 + 8y²
3x + 27x²

(5/2) - (1/e)

868/3

(e-1) / 6

sin(1) / 3

ln(17) / 4

4/3

(1 - e^-9) / 2

Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways.

type I: D = {(x, y) | 0 <= x <= 1, 0 <= y <= 1}
type II: D = {(x, y) | 0 <= y <= 1, y <= x <= 1}
1/3

Set up integrated integrals for both orders of integration. Then evaluate the double intergal using the easier order and explain why it’s easier.

type II, 9/4

(1 - cos(1)) / 2

11/3

Find the volume of the given solid.

31/8

16/3

false:
The level surface of f must be tangent to the constraint surface at an extreme value; if they intersected instead, you could keep moving along the constraint to find a higher or lower value of f. Because tangent surfaces share a perpendicular line, their gradients must be parallel (∇f = λ∇g).

With a second constraint, the path is restricted to the intersection curve of both surfaces, requiring ∇f to be perpendicular to that curve. Since ∇g and ∇h are also perpendicular to the curve, ∇f must lie in the plane spanned by them, making it a linear combination (∇f = λ∇g + μ∇h).

true or false

true

true or false

false: according to Clariaut’s Theorem, if a function has continuous second-order partial derivatives, its mixed partials myst be equal (f_xy = f_yx)

true or false

false: partial x and partial y are reversed

true

true or false

false: along “every straight" line” is not enough to guarantee that the multivariable limit exists, must be along “every single possible path”

true or false

true

true or false

false: limit might not exist at all; function may have a hole or removable discontinuity

true or false

false: ∇f(x, y) is a vector, not a scalar

true or false

true

true or false

true: the maximum and minimum of a directional derivative are ± |∇f|, respectively

true or false

false: two peaks are not necessarily separated by a valley, they are often separated by a saddle point

x = -2 + t
y = 2 - 2t
z = 4

a = sqrt(6) / 2, b = 3 sqrt(2) / 2

match the graphs of the parametric equations x = f(t) and y = g(t) in (a)-(d) with the parametric curves labeled I-IV

a) III b) I c) IV d) II

find dy/dx

find dy/dx

find an equation of the tangent to the curve at the point corresponding to the given value of the parameter

find an equation of the tangent to the curve at the point corresponding to the given value of the parameter

find an equation of the tangent to the curve at the point corresponding to the given value of the parameter

find dy/dx and d²y/dx². for which values of t is the curve concave upward?

t < 0

find dy/dx and d²y/dx². for which values of t is the curve concave upward?

find the points on the curve where the tangent is horizontal or vertical

find the points on the curve where the tangent is horizontal or vertical

find the points on the curve where the tangent is horizontal or vertical

find the exact length of the curve

find the exact length of the curve

find the exact length of the curve

find the exact length of the curve

find the distance traveled by a particle with position (x, y) as t varies in the given time interval. compare with the length of the curve

find the distance traveled by a particle with position (x, y) as t varies in the given time interval. compare with the length of the curve

e = c/a and c² = a² - b²

find the slope of th etangent line to the given polar curve at the point specified by the value of θ

find the slope of th etangent line to the given polar curve at the point specified by the value of θ

find the points on the given curve where the tangent line is horizontal or vertical

find the area of the region that is bounded by the given curve and lies in the specified sector

find the area of the region that is bounded by the given curve and lies in the specified sector

find the area enclosed by the curve

find the area enclosed by the curve

find the area of the region enclosed by one loop of the curve

find the exact length of the polar curve

find the exact length of the polar curve

find the angle between the given vector and the positive direction of the x-axis

find the magnitude of the resultant force and the angle it makes with the positive x-axis

true or false

true or false

true or false

true or false

true or false

true or false

true or false

false

true or false

true or false

true or false

find the angle between the vectors

arccos(1 / sqrt(5))

find the angle between the vectors

find the angle between the vectors

find the direction cosines and direction angles of the vector

find the direction cosines and direction angles of the vector

find the direction cosines and direction angles of the vector

find the scalar and vector projections of b onto a

find the scalar and vector projections of b onto a

find a nonzero vector orthogonal to the plane through the points P, Q, and R. find the a of triangle PQR

find a nonzero vector orthogonal to the plane through the points P, Q, and R. find the a of triangle PQR