The document contains a 24-page math exam with 6 questions testing various calculus and vector concepts. The exam asks the test-taker to solve limits of rational functions, use difference quotients to find derivatives, perform vector arithmetic and operations on matrices, analyze an electrical circuit using systems of equations, calculate properties of geometric shapes in 3D space including tetrahedrons and triangles, and apply vector calculus concepts like derivatives of position, velocity and acceleration vectors. The test-taker is instructed to show all work algebraically where possible.

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RVA/RVB (Question 1) Rational Functions
Let f(x)=
x − 4
2
2x − 16
3
(1a) (x) ?
lim
x→−∞
f =
(1b) (x) ?
lim
x→∞
f =
(1c) (x) ?
lim
x→−2−
f =
(1d) (x) ?
lim
x→−2+
f =
(1e) (x) ?
lim
x→2−
f =
(1f) (x) ?
lim
x→2+
f =
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RVA/RVB (Question 2) Difference Quotients
Let g(x) = x2 − x
(2a) Find g’(x) using the Difference Quotient.
(2b) Calculate g(x) and g’(x) when x= .
2
1
(2c) State the equation of the tangent line to g(x) at x= .
2
1
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RVA/RVB (Question 3) Difference Quotients
(3a) = ?
lim
h→0
h
(x+h) − x
2 2
(3b) = ?
lim
h→0
h
(5+h) − 25
2
(3c) = ?
lim
h→0
h
sin(x+h) − sin(x)
(3d) = ?
lim
h→0
h
sin( +h) − 1
2
π
(3e) = ?
lim
h→0
h
e − e
x+h x
(3f) = ?
lim
h→0
h
e − 1
h
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(1) Vector Arithmetic
Let ​u​ = <1,2,3>, ​v​ = <3,–2,1>, ​w​ = <4,0,6>
(1a) Find ​u​ – ​v
(1b) Find –2(​u​ – ​v​)
(1c) Find ​u​ + ​w
(1d) Find 3(​u​ + ​w​)
(1e) Find 3(​u​ + ​w​) – 2(​u​ – ​v​)
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(2) Circuit Analysis:
The currents I​1​ , I​2​ , I​3​ in a circuit with resistors R​1​, R​2​ and voltages E​1​ , E​2
are described by the following system of equations:
R​1​I​1​ + R​3​I​3​ = E​1
R​2​I​2​ + R​3​I​3​ = E​2
I​1​ + I​2​ - I​3​ = 0
Let R1 = 2 Ohms, R​2​ = 1 Ohm, R​3​ = 4 Ohms, E​1​ = 14 Volts and E​2​ = 28 Volts.
A​ = X​ = B​ =
(2a) If ​AX = B​, list the Minors of Matrix ​A​.
(2b) If ​AX = B​, state the CoFactors of the Minors of Matrix ​A​.
(2c) If ​AX = B​, find the Transpose of the CoFactors of the Minors of Matrix ​A​.
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(2) Circuit Analysis:
The currents I​1​ , I​2​ , I​3​ in a circuit with resistors R​1​, R​2​ and voltages E​1​ , E​2
are described by the following system of equations:
R​1​I​1​ + R​3​I​3​ = E​1
R​2​I​2​ + R​3​I​3​ = E​2
I​1​ + I​2​ - I​3​ = 0
Let R1 = 2 Ohms, R​2​ = 1 Ohm, R​3​ = 4 Ohms, E​1​ = 14 Volts and E​2​ = 28 Volts.
A​ = X​ = B​ =
(2d) If ​AX = B​, calculate det(​A​)​ = |A|​.
(2e) If ​AX = B​, find ​A​-1​
.
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(2) Circuit Analysis:
The currents I​1​ , I​2​ , I​3​ in a circuit with resistors R​1​, R​2​ and voltages E​1​ , E​2
are described by the following system of equations:
R​1​I​1​ + R​3​I​3​ = E​1
R​2​I​2​ + R​3​I​3​ = E​2
I​1​ + I​2​ - I​3​ = 0
Let R1 = 2 Ohms, R​2​ = 1 Ohm, R​3​ = 4 Ohms, E​1​ = 14 Volts and E​2​ = 28 Volts.
A​ = X​ = B​ =
(2f) Rewrite ​AX = B​ as ​X=A​-1​
B​.
(2g) If ​AX = B​, find the solution vector ​X​.
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(3) TetraHedrons
A molecule of Methane CH​4​ forms a Tetrahedron with Carbon at
the centroid E(k/2, k/2, k/2) and Hydrogen atoms at the 4 corners:
A(0, 0, 0), B(k, k, 0), C(k,0,k) and D(0, k, k)
(3a) Find the length of side ​CD​.
(3b) What is the angle between sides ​AC​ and ​AD​.
(3c) Calculate the bonding angle between the vectors ​ED​ and ​EA​.
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(4) r(t)​ , ​v(t)​ , ​a(t)
The motion of a particle is given by the position vector
r(t)​ = < 3cos(t), 3sin(t), t>
(4a) Find ​v(t)​ and its magnitude and interpret your result.
(4b) Find ​a(t)​ and its magnitude and interpret your result.
(4c) Calculate the dot product ​v(t)​ • ​a(t)​ and interpret your result.
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(5) Triangles In Space
Consider the triangle with vertices
A(2, 1, 0), B(1, 0, 1) and C(2, -1, 1)
(5a) Find the area of the triangle ABC.
(5b) What is the equation of the plane containing points A, B and C?
(5c) Parametrize: line parallel to the vector v = <1, 1, 1> through point S(-1, 0, 0)
(5d) Where is the point of intersection of this plane with this line?
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(6) Vector Calculus
Let ​r(t)​ be a displacement vector in space.
(6a) Find the derivative of ​r(t)​ • ​r(t)​ with respect to time.
(6b) Show that ​r(t)​ and ​v(t)​ are perpendicular when ​r(t)​ is constant.
(6c) Calculate ​r(t)​ • ​a(t)​ given ​r(t)​ is constant.
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Reference Sheet: Derivatives You Should Know Cold!
Power Functions:
x nx
d
dx
n = n−1
Trig Functions:
sin(x) os(x)
d
dx
= c cos(x) in(x)
d
dx
= − s
tan(x) (x)
d
dx
= sec2 cot(x) (x)
d
dx
= − csc2
sec(x) ec(x) tan(x)
d
dx
= s csc(x) sc(x) cot(x)
d
dx
= − c
Transcendental Functions:
e
d
dx
x = ex a n(a) a
d
dx
x = l x
ln(x)
d
dx = x
1
log (x)
d
dx a = 1
ln(a) x
1
Inverse Trig Functions:
sin (x)
d
dx
−1
= 1
√1−x2
cos (x)
d
dx
−1
= −1
√1−x2
tan (x)
d
dx
−1
= 1
1+x2 cot (x)
d
dx
−1
= −1
1+x2
Product Rule:
f(x) g(x) (x) g (x) (x) f (x)
d
dx = f ′ + g ′
Quotient Rule:
d
dx
f(x)
g(x) = g (x)
2
g(x) f (x) − f(x) g (x)
′ ′
Chain Rule:
f(g(x)) (g(x)) g (x)
d
dx = f′ ′
Difference Quotient:
f’(x) =​ lim
h→0
h
f(x+h) − f(x)
TEST1A page: 23

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