Syllabus academic test
Math
The math problems can and have to be solved exact (i.e. without using approximation techniques or a graphic calculator).
1. Functions and Graphs,
i The candidate is able to recognize compositions of standard functions. Standard funcp
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tions include polynomial functions xn , n-root functions ( n x, x n ), the power functions
ax , and its inverse the logarithm loga (x), the exponential function ex , and its inverse
the natural logarithm ln(x), trigonometric functions sin(x), and cos(x).
ii The candidate is able to analyze and draw compositions of these standard functions,
and determine domain, range, asymptotes and symmetry points or lines.
2. Algebraic Solving,
i The candidate is able to manipulate equations composed of standard functions to find
y = f (x).
ii The candidate is able to find roots of a function (f (x) = 0) using factorization techniques. The candidate is able to use the quadratic formula to find roots of quadratic
equations (ax2 + bx + c).
iii The candidate can rewrite expressions to isolate a variable and can substitute expressions into a given function.
(
ax + by = c
iv The candidate can solve systems of linear equations,
, with a, b, c, d, e, f
dx + ey = f
constants.
3. Di↵erential Calculus,
dy d
i The candidate is able to determine the first derivative (f 0 (x), dx
, dx f (x)) and second
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2
d y d
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derivative (f (x), dx2 , dx2 f (x)) of functions and use it to determine locally increasing
and decreasing behavior, extreme values, locally concave and convex behavior, and
inflection points.
ii The candidate knows the derivatives of standard functions and is able to apply the
product rule, quotient rule, and chain rule to determine derivatives.
iii The candidate is able to determine and apply derivatives to determine the tangent of a
function, to solve an optimization problem, or to solve problems concerning distance,
velocity and acceleration.
4. Integral Calculus,
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Rb
i The candidate understands the concept of integration, can calculate integralsR( a f (x)dx),
and is able to determine the antiderivative or primitive function (F (x) = f (x)dx).
The candidate knows what the constant of integration is.
ii The candidate can calculate the primitive of standard functions f (x) and functions of
the form cf (ax + b), with a, b, c constants.
iii The candidate is able to apply integration to determine surface area, volume of a solid
of revolution, arc length, mean value, and center of gravity.
5. Trigonometry,
i The candidate understands trigonometric functions sin(x), cos(x) in the context of
the unit circle. The candidate understands the terms amplitude, phase, period, and
frequency. The candidate is able to convert degrees to radians and vice-versa.
ii The candidate knows the exact values of sin(✓), cos(✓) for integer multiples of the
following angles ✓ in the first quadrant, {0, 16 ⇡, 14 ⇡, 13 ⇡, 12 ⇡}. The candidate is able to
find all solutions of equations cos(a) = cos(b), and sin(a) = sin(b).
iii The candidate knows and is able to apply the Pythagorean identity sin2 (x)+cos2 (x) =
sin(x)
1, the tangent function tan(x) = cos(x)
, and double angle formulae.
6. Geometry,
i The candidate is able to determine the surface and perimeter of two-dimensional shapes
including triangles, squares, circles, etc. The candidate is able to determine the volume
and surface area of three-dimensional figures including cubes, pyramids, cylinders,
cones, etc.
ii The candidate can use properties of lines, triangles, circles, and quadrilaterals to
determine lengths and angles. The candidate is familiar with the properties of a righttriangle, isosceles triangle, and equilateral triangle.
iii The candidate can use Pythagorean theorem, (sin, cos, tan) relations to determine
lengths/angles of right triangles. The candidate can use the law of sines and the law
of cosines to determine lengths/angles in triangles.
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Physics
1. Fundamentals,
i The candidate can use the SI base units, meter [m] for length, kilogram [kg] for mass,
second [s] for time, ampere [A] for electric current, kelvin [K] for temperature, candela
[cd] for luminous intensity, and mole [mol] for amount of substance. The candidate is
able to perform dimensional analysis to determine the dimension of physical quantities.
ii The candidate understands the concept of vector quantities (e.g. velocity ~v , acceleration ~a, force F~ , etc.) in physics, i.e. quantities that have a direction and magnitude.
The candidate can draw/add/subtract/decompose vector quantities.
iii The candidate is familiar with metric prefixes as micro- µ, milli- m, kilo- k, mega- M ,
giga- G, etc, and can transform form one to another.
2. Mechanics,
i The candidate knows relations between distance (x), velocity (v), and accelerations
(a), and knows how to analyze (x, t) , (v, t) and (a, t) diagrams. The candidate
can use relations to analyze linear motion (e.g. free fall, movement with constant
acceleration or deceleration, movement with constant friction force).
ii The candidate is able to use forces F~ and is able to apply the laws of Newton. The
candidate is familiar with gravitational forces, friction forces, tension, spring force,
etc. The candidate is able to draw these forces in a free-body-diagram and to set up a
balance of all forces. The candidate knows the equations for gravitational force, and
spring force.
~ , and is able to set up a balance of moments.
iii The candidate is able to use moments M
The candidate can calculate a moment using a given force F~ , and arm ~r. The candidate
is familiar with leverage.
iv The candidate understands principles of work W , energy E, power P and efficiency ⌘
and knows how relate these quantities to one another. The candidate is able to set up
an energy balance using gravitational energy, kinetic energy, and elastic/spring energy.
v The candidate can analyze circular motion and understand centripetal/centrifugal
force. The candidate can determine period, frequency, and angular velocity of circular
motion. The candidate can use these terms in the context of planetary orbits using a
balance of gravitational force and centripetal force.
3. Electricity,
i The candidate is familiar with the principles of electricity and understands concepts
as conduction, isolation, electrons, current I (Ampere), ion, charge Q (Coulomb),
electric potential U (Volt), resistance ⌦ [Ohm]. The candidate knows relations for
current I = Qt , potential U = QE , and Ohm’s law U = IR.
ii The candidate is able to draw and interpret simple circuit diagrams, and knows how
to apply rules of parallel and series circuits to analyse a circuit.
iii The candidate can use concepts as electric power and efficiency.
4. Thermodynamics,
i The candidate is familiar with macroscopic and microscopic view of gases, and knows
how to use the ideal gas law, pV = nRT or p = ⇢RT . The candidate is familiar with
temperature T , pressure p, volume V , Avogadro’s constant R, amount of substance n,
mass density ⇢, and molar mass M . The candidate knows definition of density ⇢ = m
V ,
R
of the specific gas constant R = M
, and of pressure p = FA .
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ii The candidate can convert temperature in Celsius to Kelvin and vice versa. The
candidate can calculate heat (thermal energy) Q = C T = cm T , with C the (molar)
heat capacity, and c the specific heat capacity (per unit mass).
5. Vibrations and Waves,
i The candidate is familiar with vibrations and corresponding terms, period T , frequency
f , amplitude, phase, resonance, and damping.
ii The candidate recognizes simple harmonic motions, and is able to analyse harmonic
motion of simple mass-spring system and of a pendulum.
iii The candidate is familiar with wave phenomena and corresponding terms, longitudinal
and transversal wave, wavelength = vT , wavespeed v, phase.
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