LAC2013 UNIT preTESTs!

Calculus AB&BC TechBook Questions Name:
Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting
Instructions
Show all work justifying each answer in the space provided.
Your answers must be supported analytically, graphically, numerically
and/or verbally! All answers must be stated in exact and simplest form if
possible.
An answer, N, that cannot be so stated, should be estimated to three
decimal places, for example:
{∀∀∀∀N | (|N|=0.001) ∩∩∩∩ (|N| < 1000)}
-.0861 = -0.086,
125.2335 = 125.234.
Any approximations, N, such that |N| ∉ [0.001, 1000) must be stated in
scientific or engineering notation rounded to three decimal places, for example:
{∀∀∀∀N | (|N|<0.001) ∪∪∪∪ (|N|=1000}
.000123 = 1.230 x 10-4 or 123.000 x 10-6,
-34567.8 = -3.457 x 10 4 or -34.568 x 10 3.
All graphs must be complete sketches including neatly labeled axes,
origin, scales, intercepts, asymptotes and function(s) or relation(s).

Copyright  2000 http://members.aol.com/calcpage
TBQ(1) A function f is defined on the closed interval from -4 to 3
and has the graph shown above.
(1a) Sketch the entire graph of y = |f(x)|.

(1b) Sketch the entire graph of y = f(|x|).

(1c) Sketch the entire graph of y = f(–x).

(1d) Sketch the entire graph of y = 2f(x + 1).

(1e) Is the graph of |f(x)| continuous at x = 1?

(1f) Does the derivative of f(–x) exist at x = 2?

TBQ(3) Let f(x) = x4 – 4x2 and g(x) = x3 – x.
(3a) Write an expression for h(x) =
f(x)
g(x).
(3b) Find all the asymptotes of h(x). Justify your answer analytically.

(3c) State the domain and range of h.
(3d) Make a complete sketch of h.

(3e) Test h(x) analytically for all three symmetries.

TBQ(5) Equal squares of side length x are removed from each
corner of a 20” by 30” piece of aluminum. The sides are turned up to
form a box with no top.
(5a) Make a drawing of the box and label each dimension in terms of x.
(5b) Write an expression for the volume V as a function of x.
(5c) Make a complete sketch of V(x). Use your sketch to show when the
volume of the box is 750 in3.

(5d) State the domain and range of V(x).
(5e) Find the maximum volume possible in this model. Describe your
procedure.
(5f) What sub domain and range make sense in the context of this model?

TBQ(7) The following implicitly defined relation describes a conic
section on the XY-plane:
f(x,y) = Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
Let: f(x,y) = 9x2 + 0xy + 4y2 – 18x + 16y – 11 = 0.
(7a1) Use factoring and completing the square to rewrite f(x,y) = 0 as:
(x – h)2
a2 +
(y – k)2
b2 = 1.
(7a2) State the following: Center Point: __________
Horizontal Vertical
Tangents: __________ Tangents: __________
Semi-Minor Semi-Major
Axis Length: __________ Axis Length: __________
Domain: __________ Range: __________
(7a3) Make a complete sketch of f(x,y) = 0 using function mode on your GC.
State y1 and y2 as explicit functions of x for the labels of your curve.

(7b1) Let B = 8, D = E = 0 and F = –36 such that
f(x,y) = 9x2 + 8xy + 4y2 – 36 = 0.
Rewrite f(x,y) = 0 in polar form as r = f(θ).
(7b2) Make a complete sketch of r = f(θ) using polar mode on your GC.
(7b3) Calculate B2 – 4AC.

(7c1) Let B = 12 such that f(x,y) = 9x2 + 12xy + 4y2 – 36 = 0.
(7c2) Make a complete sketch of r = f(θ) using polar mode on your GC.
(7c3) Calculate B2 – 4AC.

(7d1) Let B = 16 such that f(x,y) = 9x2 + 16xy + 4y2 – 36 = 0.
(7d2) Make a complete sketch of r = f(θ) using polar mode on your GC.
(7d3) Calculate B2 – 4AC.

(7e1) Make a complete sketch of 9x2 + 4y2 – 36 = 0.
(7e2) What happens when you change the values of D and E?
(7e3) Explain what the value of B2 – 4AC predicts for the behavior of a conic.

TBQ(9) Use the Difference Quotient to find the derivative of
f(x) = cos(x).
(9a) State the classic definition of the derivative as a limit.
(9b) Apply the definition of the derivative to f(x).
Show that f ’(x) =
lim
h→0 





cos(x)





cos(h) – 1
h – sin(x)





sin(h)
h .
NB: cos(α + β) = cos(α)cos(β) – sin(α)sin(β)

(9c) Make a numerical argument to find
lim
h→0
sin(h)
h
.

(9d) Make a numerical argument to find
lim
h→0
cos(h) – 1
h
.
(9e) Evaluate
lim
h→0 





cos(x)





cos(h) – 1
h – sin(x)





sin(h)
h and state f ’(x).

(9f) Find an equation, in point-slope form, of the line tangent to y = f(x) at
x =
π
2
. Restate this equation in slope-intercept form.
(9g) Find an equation, in point-slope form, of the line normal to y = f(x) at
x =
π
2. Restate this linear equation in slope-intercept form.

(9h) Make a complete sketch, ∀x ∈ [-2π, 2π], of f(x) as well as the tangent and
normal lines at x =
π
2
.
(9i) Find the area of the region enclosed by the y-axis, the tangent and the
normal. Give both the exact analytical answer and the numerical
approximation.

TBQ(11) Let f be the piece-wise defined function:
f(x) =


-x2 + C for x < 1
-2x + 5 for x ≥≥≥≥ 1
(11a) For what value of C will f be continuous at x = 1? Use the definition of
continuity at a point to justify your answer.

(11b) Use the value of C you found in part (11a) to determine whether f is
differentiable at x = 1. Justify your answer using the definition of the
derivative at a point.
(11c) Let C = 3. Determine whether f is differentiable at x = 1. Justify your
answer.

TBQ(13) Consider the relation defined implicitly by
x2 + xy + y2 = 50.
(13a) Write an expression of the form
dy
dx = f(x, y) for the slope of the curve at
any point (x, y).
(13b) Find all x and y intercepts. Write an equation of the tangent line at
each of these points.

(13c) Find all points of vertical and horizontal tangency. Write the equation
of each tangent line.

(13d) Convert the relation to polar coordinates: r = f(θ). Use polar mode to
make a complete sketch of the curve. Include all the points you found in
(b) and (c) in your sketch.

TBQ(15) On Monday, you started your career as a new employee in
the marketing division at Pepsi Cola’s Worldwide Headquarters. On
Tuesday, the VP of manufacturing asks you to redesign the 12 ounce
soda can.
(15a) Write an equation for A (the amount of aluminum used to manufacture
any cylindrical can) in terms of R (the radius of the base) and H (the
height).
(15b) Write an equation for V (the amount of soda in any cylindrical can) also
in terms of R and H.
(15c) On Wednesday, marketing research found that the most popular 12
ounce soda can has the following approximate dimensions: R = 3.068 cm
and H = 12 cm. Using this data, find the volume of the can currently in
production to the nearest cubic centimeter.

(15d) Now, manufacturing wants to save money by minimizing the aluminum
used, but marketing needs to sell the same amount of soda as they do
with the present can. Rewrite A in terms of the single variable R.
(15e) Find
dA
dR and use it to find the value of R which minimizes A. Confirm
that you have minimized A and state the new dimensions, R and H, of
the can.

(15f) A week from Thursday, a survey was conducted while test marketing
the new can on LI. One question on the survey was:
Describe the look and feel of the new can.
Do you like the new design? Why or why not?
Based on the dimensions of the can you found in (e), how do you think
most test subjects answered this question?
(15g) By next Thursday, manufacturing has implemented your new design
nationwide. You’re feeling pretty proud of yourself. You may already be
up for a promotion or even a raise! The next day, you get email from the
VP of Advertising with the subject line: “YOU’RE FIRED!!!” Explain
why advertising is so upset with you that your career at Pepsi Cola is
doomed after only 3 weeks on the job!

TBQ(17) Let f(x) = 1 – tan(x -
ππππ
2
), ∀∀∀∀x ∈∈∈∈ [0, ππππ].
(17a) State the general form of the recursive equation describing Newton’s
Method for Roots.
(17b) Apply the Quotient Rule of Differentiation to find
d
dx (tan(x)).
(17c) Apply the recursion formula to f(x).

(17d) Let x0 =
π
2
. Write an equation of the tangent line to f(x) at x0. Use this
equation to find x1.
(17e) Use prgmNEWT and Mode Fix 9 to approximate the root of f(x) nearest
to x0 by completing the following convergence table.
n xn
0 1.570796327

(17f) Find
4
3
of your best estimate to the root from part (e).
(17g) State the exact root based on part (f).
(17h) Make a complete sketch of f(x) and the tangent line at x0. Label x0, x1
and
lim
n ∞
xn on the x-axis.

TBQ(19) The volume V of a cone is increasing at the rate of 28ππππ
ft3
s
.
At the instant when the radius r of the cone is 3 ft, its volume is 12ππππ ft3
and the radius is increasing at
1
2
ft
s
.
(19a) At the instant when the radius of the cone is 3 ft, what is the
instantaneous rate of change of the area of its base?
(19b) At the instant when the radius of the cone is 3 ft, what is the
instantaneous rate of change of its height?
(19c) At the instant when the radius of the cone is 3 ft, what is the
instantaneous rate of change of the area of its base with respect to its
height.

TBQ(21) Let F(x) = ⌡⌠
0
x
sin(t2)dt , ∀∀∀∀x ∈∈∈∈ [0, 1].
(21a) Write out the trapezoidal rule with four equal subdivisions (TRAP4) on
the closed interval [0, 1] and use it to approximate F(1).
(21b) Make a complete sketch of F(1) and the trapezoids in question.

(21c) Complete the following convergence table using program lrte() and FIX4
to estimate F(1).
N LSUM RSUM TRAP
1
4
16
64
(21d) Discuss the behavior of LSUM vs. RSUM.
(21e) Discuss the behavior of TRAP.

(21f) Theorem: If f(x) is continuous, monotonic (increasing or decreasing, not
both) and positive on the closed interval [a, b], then the maximum error
in the Trapezoidal Rule approximation for the area under the graph
of f is:
|f(b) - f(a)|∆t
2 where ∆t =
(b - a)
n .
Use the given theorem to find the minimum value of n such that 3
decimal place accuracy in the TRAP column is assured.
(21g) On what interval is F increasing?
(21h) If the average rate of change of F on the closed interval [1, 3] is k, find
⌡⌠
1
3
sint(t2)dt in terms of k.

TBQ(23) Let f be a function defined ∀∀∀∀x > -5 having the following
properties:
(i) f ’’(x) =
1
3 x + 5
(ii) the line tangent to the graph of f at (4, 2) has an angle of
inclination of 45°.
Find an expression for f(x).

TBQ(25) Consider the curve y = x3 on the closed interval x ∈∈∈∈ [0,
1
2
].
(25a) Let l represent the length of the curve. Set up, but do not integrate, an
integral expression of a single variable to calculate l.
(25b) Rewrite the integral for arc length in terms of the numerical integrator
built-in to your GC.. Use your GC to estimate the arc length.

(25c) Let S represent the area of the surface generated by revolving the given
curve about the x-axis. Find S analytically by setting up and
integrating a definite integral. Your solution must include an anti-
derivative.
(25d) Rewrite the integral for surface area in terms of the numerical
integrator built-in to your GC. Use your GC to confirm your calculation
for S.

TBQ(27) A solid is constructed so that it has a circular base of
radius r centimeters and every plane section perpendicular to a
certain diameter of the base is a square, with a side of the square being
a chord of the circle.
(27a) Find the volume of the solid, V = f(r). Hint: use the relation x2 + y2 = r2.

(27b) If the solid described in part (a) expands so that the radius of the base
increases at a constant rate of
1
2
cm
min
, how fast is the volume changing
when the radius is 4cm.

TBQ(29) A particle moves on the x–axis so that its position at any
time t ≥≥≥≥ 0 is given by x(t) = 2te-t.
(29a) Find the acceleration of the particle at t = 0.
(29b) Find the velocity of the particle when the acceleration is 0.
(29c) Find the total distance traveled by the particle from t = 0 to t = 5.

TBQ(31) Let P(t) represent the number of wolves in a population at
time t years, t ≥≥≥≥ 0. The population P(t) is increasing at a rate directly
proportional to 800 – P(t), where the constant of proportionality is k.
(31a) If P(0) = 500, P(t) in terms of t and k.
(31b) If P(2) = 700, find k.
(31c) Find
lim
t→∞
P(t).

TBQ(BC33) Consider the following differential equation.
dP
dt
=
1
1 + t2 t ≥≥≥≥ 0
(BC33a) If a solution curve includes the point (0, 0), complete the table
below to estimate the value of P(1). Use Euler’s method with
∆t = 0.1 and FIX6.
t P ∆P
0.0 0.000000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

(BC33b) Refine your estimate of P(1) by completing the following
convergence table (use FIX6).
∆t P(1)
0.1
0.01
0.001
State your best estimate for P(1): _______________.

(BC33c) Solve the given differential equation analytically for P = f(t).
State the exact value of P(1): _______________.
State the error, in FIX6, of your best estimate: _______________.

TBQ(BC35) A certain rumor spreads through the community at the
rate
dy
dt
= 2y(1 – y), where y is the proportion of the population that has
heard the rumor at time t.
(BC35a) What proportion of the population has heard the rumor when it is
spreading the fastest?
(BC35b) If, at time t = 0, ten percent of the people have heard the rumor,
find y as a function of t.
(BC35c) At what time t, is the rumor spreading the fastest?

(BC35d) Graph y(t) and describe the long-term behavior of the population.
(BC35e) Graph y’(t). What does y’ tell you about the population?
(BC35f) Graph y’’(t). What does y ’’ tell you about the population?

TBQ(BC37) Let the Gamma function be defined as follows.
ΓΓΓΓ(n) = ⌡⌠
0
∞∞∞∞
tn-1e-tdt
n≥≥≥≥1, n∈∈∈∈Z
(BC37a) Find Γ(1) Numerically.

(BC37b) Find Γ(2) Analytically.

(BC37c) Find Γ(3) Numerically.

(BC37d) Find Γ(4) Analytically.

(BC37e) Restate Γ(n) as a function of n based on any pattern you have
observed in the behavior of Γ(1) .. Γ(4).

TBQ(BC39) Let R be the region enclosed by the graphs of y = e-x, x = k
with k > 0 and the coordinates axes.
(BC39a) Write an improper integral that represents the limit of the area of
the region R as k increases without bound and find the value of
the integral if it exists.
(BC39b) Find the volume, in terms of a finite k, of the solid generated if
the region R were rotated about the y–axis.

(BC39b) Find the volume, in terms of a finite k, of the solid whose base is
R and whose cross sections perpendicular to the x–axis are
squares.

TBQ(BC41) Let w(t) represent the population of worms (in millions)
and r(t) the population of robins (in thousands) on an isolated island.
A model for the interaction of these populations is given by the
following system of simultaneous, first order, ordinary differential
equations (NB: [t] = years).
dw
dt = w – wr
dr
dt = –r + wr
(BC41a) Solve for w(t) assuming the robins have died out.
(BC41b) Describe the population growth for worms in the absence of
robins: (circle one)
(1) Exponential Growth
(2) Exponential Decay
(3) Heating Curve
(4) Cooling Curve
(5) Logistic Growth
(6) Logistic Decay
(7) Stable Equilibrium
(8) Unstable Equilibrium

(BC41c) Solve for r(t) assuming the worms have died out.
(BC41d) Describe the population growth for robins in the absence of
worms: (circle one)
(3) Heating Curve
(4) Cooling Curve
(5) Logistic Growth
(6) Logistic Decay
(BC41e) Find all Equilibrium points for the interacting populations.

(BC41f) Using a Slope Field for
dr
dw
in the window [0, 4]x[0, 4] to plot a
trajectory for the initial populations of 2x106 worms and 2x103 robins. Use
your trajectory to estimate the minimum and maximum populations for both
species.
Wmin = _______________
Wmax= _______________
Rmin = _______________
Rmax = _______________

(BC41g) Using a Slope Field for
dr
dw
in the window [0, 4]x[0, 4] to plot a
trajectory for the initial populations of 2x106 worms and 3x103 robins. Use
your trajectory to estimate the minimum and maximum populations for both
species.
Wmin = _______________
Wmax= _______________
Rmin = _______________
Rmax = _______________

(BC41h) Describe the interaction of these two species: (circle one)
(1) Symbiotic
(2) Competitive
(3) Predator Prey
(4) Non-interactive
(BC41i) The people on this island do not usually interact with the robins
or the worms. However, they do love robins! Does it make sense
to introduce the additional 1000 robins at t = 0? Please explain
why or why not in paragraph form.

TBQ(BC43) Let P(t) represent population of fish in one pond of a fish
farm. The following differential equation models the growth of the fish
population when 75 fish are harvested at the end of each year to be sold
to the fish market (NB: [t] = years).
dP
dt = 2P – 0.01P2 – 75
(BC43a) When is
dP
dt
= 0?
(BC43b) Describe the population growth for these fish: (circle one)
(3) Heating Curve
(4) Cooling Curve
(5) Logistic Growth
(6) Logistic Decay
(BC43c) Find the general solution for the differential equation: P = f(t).

(BC43d) Given the initial condition P(0) = 40, find the particular solution
to the given differential equation: P = f(t). Graph your solution.
Explain what this graph predicts for the future of your business.

(BC43e) Given the initial condition P(0) = 60, find the particular solution

(BC43f) Given the initial condition P(0) = 140, find the particular solution

(BC43g) Given the initial condition P(0) = 160, find the particular solution

TBQ(BC45) Consider the series ∑
n=2
∞∞∞∞
1
npln(n)
(BC45a) Show that this series converges when p > 1.
Use the Comparison Test.

(BC45b) Does this series converge for p = 1?
Use the Integral Test.

(BC45c) Show divergence when 0 ≤ p < 1!
Choose a convergence test.

TBQ(BC47) Let f(x) = x –
x2
2
+
x3
3
– … = ∑
n=1
∞∞∞∞
(-1)n-1xn
n
(BC47a) Show that the series for f(1) converges.
(BC47b) Estimate f(1).

TBQ(BC49) Let f(x) =
x
1 + x
(BC49a) Generate the first six non–zero terms, and the general term, of
the Taylor Series Expansion for f(x) near x = 0. Find the interval
of convergence.
(BC49b) Generate the first six non–zero terms, and the general term, of
the Taylor Series Expansion for ⌡⌠f(x)dx near x = 0. Find the
interval of convergence.

(BC49c) Use the first five terms of the antiderivative to estimate ⌡⌠
0
1
4
f(x)dx
(use FIX12).
(BC49d) Find the error in your approximation (use FIX12). How
accurately can you state the answer?

TBQ(BC51) Consider the area: A = ⌡⌠
0
1
sin(x2)dx.
(BC51a) Write the first four non–zero terms of the MacLaurin Expansion
for sin(x2) and ⌡⌠sin(x2)dx. What are the general terms?
(BC51b) Use the first three non–zero terms of your expansion for the
antiderivative to estimate ⌡⌠
0
1
sin(x2)dx (use FIX12).
(BC51c) Find the error in your approximation (use FIX12). How
accurately can you state the answer?

TBQ(BC53) A particle moves in the XY-plane so that at any time t ≥≥≥≥ 0 its
position (x, y) is given by x(t) = e t + e -t and y(t) = e t – e -t.
(BC53a) Find the velocity vector for any t ≥ 0.
(BC53b) Find
lim
t → ∞
dy
dt
dx
dt
.

(BC53c) The particle moves on a hyperbola. Find an equation for this
hyperbola in terms of x and y.
(BC53d) Sketch the path of the particle showing the velocity vector and
initial position when t = 0.

TBQ(BC55) Consider the curves defined by r1 = 3cosθθθθ and r2 = 1 + cosθθθθ.
(BC55a) Sketch the curves and shade–in the region inside the curve r1 and
outside the curve r2.
(BC55b) Find the area of the shaded region by setting up and evaluating a
definite integral. Your work must include an anti–derivative and
how you arrived at it.

TBQ(BC41) Given
d2y
dx2 = xy, y(0) = 1, y ’(0) = -1
(BC41a) Use a MacLaurin series to find a sixth degree Taylor Polynomial
solving the given second order ordinary differential equation.

(BC41b) Let P4(x) be the fourth degree Taylor Polynomial approzimating
y=f(x). Find the roots of y = P4(x).
(BC41c) Let P4(x) be the fourth degree Taylor Polynomial approzimating
y=f(x). Find the absolute max of y = P4(x).

(BC41d) Complete the following table and make a sketch of y=P4(x).
x P4(x) |P4(x) – f(x)|
-2
- 2
-1
-1
2
0
1
2
1
2
2

TBQ(BC43) The equation for the charge Q(t) on a capacitor in a circuit
with inductance L, capacitance C and resistance R satisfies the
differential equation:
L
d2Q
dt2 + RL
dQ
dt +
Q
C = 0.
NB: [Q, L, R, C, t] = Coulomb, Henry, Ohm, Farad, and second
(BC43a) Given L=1, R=2 and C=4, use a characteristic equation to find the
general solution for Q(t).

(BC43b) Given L=1, R=1 and C=4, use a characteristic equation to find the

(BC43c) Given L=8, R=2 and C=4, use a characteristic equation to find the

(BC43d) How did reducing the resistance of the circuit affect the
accumulated charge on the capacitor? Compare the results from
parts a and b.

(BC43e) How did increasing the inductance of the circuit affect the
accumulated charge on the capacitor? Compare the results from
parts a and c.

LAC2013 UNIT preTESTs!

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