Calculus III Midterm Study Guide

Consider the following vector equation. r(t)= <t^6, t^5> (a) Find r'(t) (b) Sketch the plane curve together with the position vector r(t) and the tangent vector r ′(t) for the given value of t = 1.

(a) <6t^5, 5t^4> (b) opening to the right, intersection at x=1

Consider the following vector function. r(t)= <4sin(t), t, 4cos(t)> (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use the formula K(t)= |T'(t)|/|r'(t)| to find the curvature

(a) T(t)= <4cos(t)/sqrt17, 1/sqrt17, -4sin(t)/sqrt17> N(t)= <-sin(t), 0, -cos(t)> (b) 4/17

Find an equation of the sphere that passes through the point (1, 8, 5) and has center (3, 5, −1).

(x-3)^2 + (y-5)^2 + (z+1)^2 = 49

Find a * b a= <6, -3, 2> b= <3, 6, -1>

-2

Find an equation of the plane. the plane through the point (6, −2, 6) and perpendicular to the vector-i + 4j + 5k

-x + 4y + 5z = 16

Sketch the region bounded by the paraboloids z = x^2 + y^2 and z= 8 - x^2 -y^2 (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) -(Write an equation for the cross section at z = 1 using x and y.) -(Write an equation for the cross section at z = 2 using x and y.) -(Write an equation for the cross section at z = 4 using x and y.) -(Write an equation for the cross section at z = 6 using x and y.) -(Write an equation for the cross section at z = 7 using x and y.)

1 = x^2 + y^2 2 = x^2 + y^2 4 = x^2 +y^2 2 = x^2 +y^2 1 = x^2 + y^2

A manufacturer has modeled its yearly production function P (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function P(L, K)= 1.47L^0.65K^0.35 where L is the number of labor hours (in thousands) and K is the invested capital (in millions of dollars). Find P(125, 30) and interpret it. (Round your answers to one decimal place.) P(125, 30)= ________ , so when the manufacturer invests $________million in capital and ________ thousand hours of labor are completed yearly, the monetary value of the production is about $________ million

111.5 30 125 111.5

Evaluate the integral. integral 0 to 2 (8ti - t^3j + 2t^3k) dt

16i -4j + 8k

Find an equation of the set of all points equidistant from the points A(−3, 6, 3) and B(5, 1, −1). Describe the set.

16x - 10y - 8z = -27 a plane perpendicular to AB

Find the length of the curve. r(t) = <4t, t^2, 1/6t^3> 0, < t < 1

25/6

Use Lagrange multipliers to find three positive numbers whose sum is 290 and whose product is maximum. (Enter your answers as a comma-separated list.)

290/3, 290/3, 290/3

Find all the second partial derivatives. f(x, y)= x^6y-2x^3y^2 fxx(x, y) fxy(x, y) fyx(x, y) fyy(x, y)

30x^4y - 12xy^2 6x^5 - 12x^2y 6x^5 - 12x^2y -4x^3

The radius of a right circular cone is increasing at a rate of 1.6 in/s while its height is decreasing at a rate of 2.6 in/s. At what rate is the volume of the cone changing when the radius is 110 in. and the height is 119 in.?

3476pi in^3/s

Explain why the function is differentiable at the given point. f(x, y)= x^3y^2, (-3, 1) The partial derivatives are fx(x, y)= ________ and fy(x, y)= ________, so fx(-3, 1)= _______ and fy(-3, 1)= ________. Both fx and fy are continuous functions so, f is differentiable at (-3, 1) Find the linearization L(x, y) of the function at (−3, 1).

3x^2y^2 2x^3y^2 27 -54 L(x, y)= 27x - 54y + 108

Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim (x, y)->(1, 2) (8x^3 - x^2y^2)

4

Find an equation of the plane. the plane that contains the line x = 2 + t, y = 3 − t, z = 3 − 3t and is parallel to the plane 5x + 2y + z = 3

5x + 2y + z = 19

A tow truck drags a stalled car along a road. The chain makes an angle of 30° with the road and the tension in the chain is 1,200 N. How much work (in J) is done by the truck in pulling the car 1 km?

600000sqrt3

Find the area of the parallelogram with vertices A(−3, 0), B(−1, 7), C(9, 6), and D(7, −1).

72

Find the cross product a x b a= 6i + 6j - 6k b= 6i - 6j +6k Verify that it is orthogonal to both a and b. (a x b) * a = (a x b) * b =

<0, -72, -72> 0 0

Find the sum of the given vectors. a=<1, 5, -3> b=<0, 0, 8> Illustrate geometrically.

<1, 5, 5> a starts at (x, y, z) = (0, 0, 0) and ends at (x, y, z) = (1, 5, -3) b starts at (x, y, z) = (1, 5, -3) and ends at (x, y, z) = (1, 5, 5) a + b starts at (x, y, z) = (0, 0, 0) and ends at (x, y, z) = (1, 5, 5)

Find a unit vector that has the same direction as the given vector. 2i - j + 2k

<2/3, -1/3, 2/3>

Find the directional derivative of f at the given point in the direction indicated by the angle 𝜃. f(x, y)= xy^3 - x^2, (1, 3), 𝜃=pi/3

Duf(1, 3)= (25+27sqrt3)/2

Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y, z)= x^2y + y^2z, (2, 4, 6), v= <2, -1, 2>

Duf(2, 4, 6)= 4

Determine whether the planes are parallel, perpendicular, or neither. 5x + 20y - 15z = 1, -27x + 54y + 63z = 0 If neither, find the angle between them. (Use degrees and round to one decimal place. If the planes are parallel or perpendicular, enter PARALLEL or PERPENDICULAR, respectively.)

perpendicular PERPENDICULAR

Sketch the graph of the function. f(x, y)= 14 - 4x - 5y

pyramid

Find the derivative, r'(t), of the vector function. r(t)= <sqrt(t-5), 9, 4/t^2>

r'(t)= <1/2sqrt(t-5), 0, -8/t^3>

A force with magnitude 15 N acts directly upward from the xy-plane on an object with mass 3 kg. The object starts at the origin with initial velocity v(0)= i - j Find its position function and its speed at time t.

r(t)= ti - tj + 5/2t^2k |v(t)|= sqrt(2+25t^2)

Find the points on the cone z^2= x^2 + y^2 that are closest to the point (8, 2, 0).

smaller z-value (x, y, z)= (4, 1, -sqrt17) larger z-value (x, y, z)= (4, 1, sqrt17)

A ship is sailing west at a speed of 38 km/h and a dog is running due north on the deck of the ship at 6 km/h. Find the speed (in km/h) and direction of the dog relative to the surface of the water. (Round your answers to one decimal place.)

speed: 38.5 km/h direction: 81.0 degrees W

Find the velocity, acceleration, and speed of a particle with the given position function. r(t)= <6cos(t), 7t, 6sin(t)>

v(t)= <-6sin(t), 7, 6cos(t)> a(t)= <-6cos(t), 0, -6sin(t)> |v(t)|= sqrt85

Find the velocity, acceleration, and speed of a particle with the given position function. r(t)= e^ti + e^4tj Sketch the path of the particle and draw the velocity and acceleration vectors for t = 0.

v(t)= e^ti + 4e^4tj a(t)= e^ti + 16e^4tj |v(t)|= sqrt(e^2ti + 16e^8tj) Both arrow pointing to the right

Find a vector equation and parametric equations for the line segment that joins P to Q. P(3.5, -1.2, 2.1) Q(1.8, 0.3, 2.1) vector equation parametric equations

vector: r(t)= <3.5-1.7t, -1.2+1.5t, 2.1> parametric equation: (3.5-1.7t, -1.2+1.5t, 2.1)

Match the equation with its graph. y= 7x^2 - 7z^2

vertical parabola (PS5 shape)

A contour map of a function is shown. Use it to make a rough sketch of the graph of f. (Bulls eye graph)

vertical trumpet horn shape

Consider the following. 2(x-3)^2 + (y-5)^2 + (z-7)^2 = 10, (4, 7, 9) (a) Find an equation of the tangent plane to the given surface at the specified point. (b) Find an equation of the normal line to the given surface at the specified point. (x(t), y(t), z(t))=

x + y + z= 20 (4+t, 7+t, 9+t)

Which coordinate plane is closest to the point (8, 4, 7)? Find an equation of the sphere with center (8, 4, 7) that just touches (at one point) that coordinate plane.?

xz-plane (x-8)^2 + (y-4)^2 + (z-7)^2 = 16

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t)= <3sin(t), 5cos(t)>

y-axis= -5, 5 x-axis -3, 3 the circle moves clockwise

Find an equation of the plane that contains the curve with the given vector equation. r(t)= <t, 5, t^2>

y=5

Find an equation of the tangent plane to the given surface at the specified point. z= ln(x-9y), (10, 1, 0)

z= x - 9y - 1

Find | u x v |, |u|= 2, |v|= 7 degrees: 60 Determine whether u ⨯ v is directed into the screen or out of the screen.

| u x v |= 7sqrt3 u ⨯ v is directed into the screen.

Use the chain rule to find ∂z/∂s and ∂z/∂t .z= (x-y)^9, x=s^2t, y=st^2 ∂z/∂s= ∂z/∂t=

∂z/∂s= (9(x-y)^8)(2st-t^2) ∂z/∂t= (9(x-y)^8)(s^2-2st)

Use the equations ∂z/∂x= -(∂F/∂x)/(∂F/∂z) and ∂z/∂y= -(∂F/∂y)/(∂F/∂z) to find ∂z/∂x and ∂z/∂y x^2 + 8y^2 + 3z^2 = 1

∂z/∂x=-x/3z ∂z/∂y=-8y/3z

Use the formula K(t)= |f''(x)|/[1+f'(x))^2]^3/2 to find the curvature y=5x^4

K(x)= 60x^2/(1+400x^6)^3/2

Find the differential of the function. H= x^7y^8 + y^6z^3 dH=

dH= (7x^6y^8)dx + (8x^7y^7 + 6y^5z^3)dy + (3y^6z^2)dz

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a= <-1, 6, 5> b=<3, 4, 1>

exact: cos^-1 (26/(sqrt 1612) approximate: 50 degrees

Find the first partial derivatives of the function. w= e^v/u+v^3

fu(u,v)= -e^v/(u+v^3)^2 fv(u,v)= e^v(u+v^3-3v^2)/(u+v^3)^2

Find the first partial derivatives of the function. z= xsin(xy)

fx(x,y)= xycos(xy)+sin(xy) fy(x,y)= x^2cos(xy)

Match the equation with its graph. x^2 + 25y^2 + 36z^2 = 1

horizontal ellipsoid

Match the equation with its graph. x^2 - y^2 + z^2 = 4

horizontal wormhole shape

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) f(x, y)= 5 - x^4 + 2x^2 - y^2

local maximum value(s): 6 local minimum value(s): DNE saddle point(s) (x, y)= (0, 0)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) f(x, y)= x^2 + xy + y^2 + 5y

local maximum value(s): DNE local minimum value(s): -25/3 saddle point(s) (x, y)= DNE

This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z)= 8x + 8y + 5z, 4x^2 +4y^2 +5z^2 =37

maximum value: 37 minimum value: -37

This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y)= x^2 - y^2, x^2 + y^2= 9

maximum value: 9 minimum value: -9