Quiz 1math.fau.edu/markus/courses/calculus3-quiz1-12.pdf · 2019-11-15 · 2 Calculus III, Quiz 1, August 23, 2019 2. An airplane traveling horizonally at 100 meter per second at

Department of Mathematical SciencesInstructor: Markus SchmidmeierCalculus IIIAugust 23, 2019 Name:

Quiz 1

1. For each of the parametric equations below, make a sketch ofthe curve and determine the equation of the tangent line at thepoint where the parameter has value t = 0.

(a) x = cos(t), y = sin(t)

(b) x = et, y = t

2 Calculus III, Quiz 1, August 23, 2019

2. An airplane traveling horizonally at 100 meter per second at anelevation of 4500 meters must drop an emergency package on atarget on the ground. The trajectory of the package is given by

x = 100 t, y = −5 t2 + 4500, t ≥ 0

where the origin is the point on the ground directly beneath theplane at the moment of release. How many meters before thetarget should the package be released in order to hit the target?

Department of Mathematical SciencesInstructor: Markus SchmidmeierCalculus IIISeptember 6, 2019 Name:

Quiz 2

1. Consider the curve given by the parametric equation

r = cos(3ϑ).

(a) Make a sketch of the curve.

(b) Determine a definite integral that represents the area of theregion enclosed by one petal.

(c) Evaluate the integral in (b).

2 Calculus III, Quiz 2, September 6, 2019

2. Consider the curve given by the parametric equation

r = 2 cos(ϑ).

(a) Make a sketch of the curve.

(b) Determine a definite integral that represents the length of thearc of the curve on the interval 0 ≤ ϑ ≤ π

2.

(c) Evaluate the integral in (b).

(d) Use a familiar formula in geometry to confirm the answer in (c).


Quiz 3

1. Recall that ~ı = 〈1, 0, 0〉, ~ = 〈0, 1, 0〉, ~k = 〈0, 0, 1〉.

(a) Find the dot product of the vectors 〈1, 2, 3〉 and 〈−1, 2,−1〉 andthe angle between them.

(b) Find the angle between the vectors ~ı− ~ and~~k − .

(c) Find three unit vectors which are perpendicular (i.e. orthogo-nal) to ~ı− ~.


2. The point P : (0, 1,√3) is given in cartesian coordinates.

(a) Find the cylindrical coordinates for P .

(b) Find the spherical coordinates for P .

(c) The equation of a surface in spherical coordinates is given bythe equation

ϕ =π

6.

Identify and graph the surface!


Quiz 4

1. Recall that ~ı = 〈1, 0, 0〉, ~ = 〈0, 1, 0〉, ~k = 〈0, 0, 1〉.

(a) Find the dot product of the vectors 〈1, 2, 3〉 and 〈−1, 2,−1〉 andthe angle between them.

(b) Find the angle between the vectors ~ı− ~ and~~k − .

(c) Find three unit vectors which are perpendicular (i.e. orthogo-nal) to ~ı− ~.


2. The point P : (0, 1,√3) is given in cartesian coordinates.

(a) Find the cylindrical coordinates for P .

(b) Find the spherical coordinates for P .

(c) The equation of a surface in spherical coordinates is given bythe equation

ϕ =π

6.

Identify and graph the surface!


Quiz 5

1. Consider the two planes given by the equations

P1 : x + y = 1, P2 : x + z = 1.

Decide if the planes are parallel or intersecting. If they areparallel, find the distance! If they are intersecting, find theparametric equations of the line of intersection, and the anglebetween the two planes.


2. A solar panel is mounted on the roof of a house. The cornershave the following coordinates, measured in meters.

A : (2, 0, 0), B : (0, 0, 2), C : (2, 6, 0), D : (0, 6, 2)

(a) Determine the equation of the plane given by points A,B,C,and show that point D is on that plane.

(b) What is the shape of the panel? (Is it a parallelogram?) Findthe area!

(c) Find parametric equations for the line through the midpoint ofthe panel, and perpendicular to the plane of the panel.

(d) Suppose the sun is in direction ~u = 〈1, 2, 2〉. Determine theangle ϑ between ~u and the direction of the line in (c).

(e) Suppose the power output of the panel is

100W/m2 · (area in m2) · cos(ϑ)2.

Estimate the power output using the area from (b) and theangle from (d)!

Department of Mathematical SciencesInstructor: Markus SchmidmeierCalculus IIIOctober 4, 2019 Name:

Quiz 6

1. Consider the car on the race trac given by position vector

~r(t) = 〈2 cos(t), sin(2t)〉.

(a) Find the velocity and accelleration vectors at any time t.

(b) At time t = π4, determine the velocity, the speed, the accellera-

tion, and the unit tangent vector.

(c) Make a sketch of the race track and indictate the vectors in (b)in your sketch.

5 best quizzes: average: letter grade:

2 Calculus III, Quiz 6, October 4, 2019

2. The trajectory of an object is a circular helix given by the equa-tion

~r(t) = 〈4 cos(t), 4 sin(t), 3t〉.

(a) Find the arc length of the curve on the interval 0 ≤ t ≤ 2π.

(b) Find the curvature κ at t = 0. What is the radius of thecorresponding osculating circle?


Quiz 7

1. A cannon is fired from a cliff of height 100m above the groundwith an initial velocity of 100 m/sec and at an angle of 30◦ abovethe horizontal. You may estimate gravity at g = 10 m/sec2.

(a) Determine the position ~r(t) of the projectile at any time t.

(b) Find the maximum height above ground of the projectile.


2. An object is moving on an elliptical orbit given by positionvector

~r(t) = 〈a cos(ωt), b sin(ωt)〉.

(a) Find the tangential and normal components of the accellerationvector at time t = 0.

(b) Find the curvature κ of the orbit at time t = 0, and determinethe corresponding radius R of the osculating circle.

(c) Make a sketch of the orbit if a = 2, b = 1, ω = 1 and indicatethe osculating circle in your sketch.


Quiz 8

1. Consider the function in two variables,

f(x, y) = y2.

(a) Find the level curves for c = 4 and c = 9. (b) Make a sketchof these level curves. (c) In words, describe the shape of thesurface.


2. Consider the surface given by

h(x, y) = x2 − 2xy + y2.

(a) Find the equation of the tangent plane to the surface at thepoint P : (1, 2).

(b) Use the tangent plane to estimate h(0.9, 2.1).

(c) Compute the actual value of h(0.9, 2.1) and determine the errorin your estimate in (b).


Quiz 9

1. (a) Finddz

dtusing the chain rule.

z = 3 cos(x)− sin(xy), x =1

t, y = 3t

(b) The radius of a right circular cylinder is increasing at a rate of2 cm/min whereas the height of the cylinder is decreasing at arate of 3 cm/min. Find the rate of change of the volume of thecylinder when the radius is 10 cm and the height is 20 cm.


2. Consider the function in two variables

f(x, y) = 36− 4x2 − 9y2

and the point P : (1,−1).

(a) Find the gradient of f(x, y) at P .

(b) Find the directional derivative of f(x, y) at P in the direction~u = 3

5~ı+ 4

5~.

(c) At P , in which direction does f(x, y) increase most rapidly? inwhich direction does f(x, y) decrease most rapidly? in whichdirection is the level curve?

Department of Mathematical SciencesInstructor: Markus SchmidmeierCalculus IIINovember 1, 2019 Name:

Quiz 10

1. Find all critical points and use the second derivative test todecide for each critical point if it is a maximum, a minimum, asaddle point or none of these.

f(x, y) = −x3 + 4xy − 2y2 + 1

Decide for each maximum or minimum if it is local and/ orglobal.

2 Calculus III, Quiz 10, November 1, 2019

2. Find the position of the global maximum and the global mini-mum, and the maximum and minimum values for the function

f(x, y) = x2 + y2 − 2y + 1

on the region R = {(x, y) : x2 + y2 = 4}.


Quiz 11

1. A rectangular box without a top is to be made from 48 ft2

of cardboard. Find the maximum volume of such a box, andspecify the dimensions (= lengths of the sides) of the box.


2. In each of the following problems, make a sketch of the regionR, convert the double integral into an iterated integral, andevaluate the iterated integral.

(a)∫∫

Rsin(x) cos(3y) dA, R = [0, π]× [−π

6, π6]

(b)∫∫

Rx ex+3y dA, R = [0, 1]× [1, 2]


Quiz 12

1. Consider the region R in the first quadrant bounded by thex-axis, the y-axis and the line y = 1 − x.

(a) Make a sketch of the region R and decide if the region is type I,type II, both or neither.

(b) Convert the double integral into an iterated integral.∫∫R

ex dA

(c) Evaluate the iterated integral in (b).


2. Consider the four-leaved rose given by the equation in polarcoordinates.

r = cos(2ϑ)

(a) Make a sketch.

(b) Let R be the region bounded by the x-axis and the part ofthe four-leaved rose in the first quadrant (where 0 ≤ ϑ ≤ π

4).

Convert the double integral∫∫R

r dA

into an iterated integral (note the extra r).

(c) Evaluate the iterated integral in part (b).

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Quiz 1math.fau.edu/markus/courses/calculus3-quiz1-12.pdf · 2019-11-15 · 2 Calculus III, Quiz 1, August 23, 2019 2. An airplane traveling horizonally at 100 meter per second at