3. MAT225 TEST1A Name:
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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
TEST1A page: 3
9. MAT225 TEST1A Name:
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(2) Circuit Analysis:
The currents I1 , I2 , I3 in a circuit with resistors R1, R2 and voltages E1 , E2
are described by the following system of equations:
R1I1 + R3I3 = E1
R2I2 + R3I3 = E2
I1 + I2 - I3 = 0
Let R1 = 2 Ohms, R2 = 1 Ohm, R3 = 4 Ohms, E1 = 14 Volts and E2 = 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.
TEST1A page: 9
15. MAT225 TEST1A Name:
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(3) TetraHedrons
A molecule of Methane CH4 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.
TEST1A page: 15
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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.
TEST1A page: 17
19. MAT225 TEST1A Name:
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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)?
TEST1A page: 19
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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.
TEST1A page: 21
23. MAT225 TEST1A Name:
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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