preTEST1A Solved Multivariable Calculus

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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) = x​2​
- 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
(1a) = ?
lim
h→0
h
(x+h) − x
2 2
(1b) = ?
lim
h→0
h
(5+h) − 25
2
(1c) = ?
lim
h→0
h
sin(x+h) − sin(x)
(1d) = ?
lim
h→0
h
sin( +h) − 1
2
π
(1e) = ?
lim
h→0
h
e − e
x+h x
(1f) = ?
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​ + ​v​) – 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) Where is the point of intersection of this plane with the line parallel to
the vector v = <1, 1, 1> passing through the point S(-1, 0, 0)?
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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)
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