Calculus
What is the derivative of f(x)=x2?
2x
Using the power rule, bring the exponent down as a coefficient and subtract 1 from the exponent: d(x2)/dx = 2x1 = 2x
Which ancient Greek mathematician is famously known for writing Elements, one of the most influential mathematical textbooks in history?
Euclid
Often called the Father of Geometry, organized mathematical knowledge into definitions, axioms, and logical proofs in his book Elements.
Solve for x: 2x + 6 = 14.
x = 4
Subtract 6 from both sides: 2x = 8.
Divide both sides by 2: x = 4.
What is the smallest prime number?
2
A prime number is a positive integer greater than 1 that has exactly two positive factors: 1 and itself.
What is the mean (average) of the numbers 2, 4, 6, 8, and 10?
6
Add all the values and divide by the number of observations:
(2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6.
What is the limit as x approaches 2 of (3x + 4)?
10
Substitute x = 2 into the expression: 3(2) + 4 = 6 + 4 = 10.
What mathematician is credited with developing the foundations of calculus independently of Isaac Newton?
Gottfried Wilhelm Leibniz
Leibniz and Newton independently developed calculus during the 17th century. Leibniz introduced much of the notation still used today, including the integral symbol and differential notation.
Factor the following quadratic expression: x2 + 7x + 12.
(x + 3)(x + 4)
Find two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4.
If you flip a fair coin three times, how many possible sequences of heads and tails can you get?
8 possible sequences
Each coin flip has 2 possible outcomes. Therefore, the total number of sequences is:
23 = 8.
What is the median of the following data set: 3, 7, 9, 12, 15?
9
The median is the middle value when the data is arranged in numerical order.
What is the derivative of f(x) = sin(x2)?
2x cos(x2)
Use the chain rule. The derivative of the outer function is cos(x2), and the derivative of the inner function x2 is 2x. Multiply them together to get 2x cos(x2).
Which mathematician developed a famous formula connecting five fundamental mathematical constants: e, i, pi, 1, and 0?
Leonhard Euler
Euler's identity is e(i*pi) + 1 = 0. It connects five important mathematical constants in one elegant equation.
If f(x) = 2x2 - 3x + 1, what is f(-2)?
15
Substitute -2 for x:
2(-2)2 - 3(-2) + 1
= 8 + 6 + 1
= 15.
In a group of 10 people, how many different ways can you select a committee of 3 people if order does not matter?
120 different ways
Use the combination formula:
10 choose 3 = 10! / (3! x 7!)
= (10 x 9 x 8) / (3 x 2 x 1)
= 720 / 6
= 120.
A fair six-sided die is rolled once. What is the probability of rolling a number greater than 4?
1/3 or approximately 33.3%
The favorable outcomes are 5 and 6. There are 2 favorable outcomes out of 6 possible outcomes.
Probability = 2/6 = 1/3 = 33.3%.
What is the integral from 0 to 2 of (3x2 + 2x) dx?
12
Find the antiderivative: x3 + x2. Evaluate at the boundaries:
(23 + 22) - (03 + 02) = 8 + 4 = 12.
Which mathematician, working in ancient India, is credited with one of the earliest systematic treatments of zero as a number and rules for arithmetic involving zero?
Brahmagupta
In the 7th century, Brahmagupta described arithmetic rules involving zero and negative numbers in his work Brahmasphutasiddhanta. Earlier civilizations had used zero as a placeholder, but his work helped establish zero's role in arithmetic.
Solve the following system of equations:
2x + y = 11
x - y = 1
x = 4 and y = 3
Add the two equations together to eliminate y:
3x = 12
Divide by 3: x = 4.
Substitute x = 4 into the second equation:
4 - y = 1
Therefore, y = 3.
A connected planar graph has 8 vertices and 12 edges. Using Euler's formula, how many faces does it have?
6 faces
Euler's formula is:
V - E + F = 2
Substitute the given values:
8 - 12 + F = 2
F = 6.
A data set has a mean of 50 and a standard deviation of 5. What is the z-score of an observation with a value of 60?
2
Use the z-score formula:
z = (x - mean) / standard deviation
Substitute the values:
z = (60 - 50) / 5
z = 10 / 5
z = 2
A particle moves along a straight line with velocity v(t) = 3t2 - 12t + 9. At what times is the particle momentarily at rest?
t = 1 and t = 3
A particle is momentarily at rest when its velocity equals zero.
Set the equation equal to zero:
3t2 - 12t + 9 = 0
Divide everything by 3:
t2 - 4t + 3 = 0
Factor:
(t - 1)(t - 3) = 0
Therefore, t = 1 and t = 3.
Which mathematician proved that no three positive integers a, b, and c can satisfy an + bn = cn for any integer n greater than 2?
Andrew Wiles
This is Fermat's Last Theorem, proposed by Pierre de Fermat. Andrew Wiles proved it in the 1990s using advanced mathematical techniques involving elliptic curves and modular forms.
Solve the equation: x4 - 5x2 + 4 = 0.
x = -2, -1, 1, and 2.
Substitute u = x2.
The equation becomes:
u2 - 5u + 4 = 0.
Factor:
(u - 1)(u - 4) = 0.
Therefore:
x2 = 1 or x2 = 4.
Taking the positive and negative square roots gives:
x = -2, -1, 1, and 2.
How many edges does a complete graph with 10 vertices have?
45 edges
Use the complete graph formula:
E = n(n - 1) / 2
Substitute n = 10:
E = 10(10 - 1) / 2
E = 10(9) / 2
E = 45.
A researcher conducts a hypothesis test and obtains a p-value of 0.03. If the significance level is 0.05, what is the correct statistical decision?
Reject the null hypothesis
Since the p-value (0.03) is smaller than the significance level (0.05), the result is statistically significant at the 5% level. Therefore, we reject the null hypothesis.