CESS & Examen d'entrée (Belgium) · Mathematics

Differential and Integral Calculus, Differential Equations, and Probability

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Compute the derivative $f^{'} (x)$ of the rational function $f (x) = \frac{3 x - 1}{2 x + 5}$ .
Find the derivative $\frac{d y}{d x}$ of $y = ln (x^{2} + 4)$ .
Determine the local minimum value of the polynomial function $f (x) = 2 x^{3} - 3 x^{2} - 12 x + 5$ .
Find the $x$ -coordinate of the point of inflection of the curve $y = x^{3} - 6 x^{2} + 9 x + 2$ .
Evaluate the definite integral $\int_{1}^{3} (3 x^{2} - 4 x + 1) d x$ .
Calculate $\int 2 x x^{2} + 5 d x$ .
Evaluate $\int x e^{2 x} d x$ .
Find the area of the region enclosed between the parabola $y = x^{2}$ and the line $y = 2 x$ .
Evaluate the limit $lim_{x \to 0} \frac{s i n ( 3 x )}{e ^{2 x} - 1}$ .
Calculate $lim_{x \to \infty} \frac{5 x ^{3} - 2 x + 7}{2 x ^{3} + 4 x ^{2} - 1}$ .
Find the general solution to the differential equation $\frac{d y}{d x} = 3 x^{2} y$ for $y > 0$ .
Solve the initial value problem $\frac{d y}{d x} = 4 x + 1$ with initial condition $y (1) = 5$ .
A farmer encloses a rectangular field against a straight stone wall (using the wall as one side) with 120 meters of fencing. What is the maximum possible area of the field?
A random variable $X$ is normally distributed with mean $μ = 50$ and standard deviation $σ = 8$ . Compute the standard $z$ -score corresponding to $x = 66$ .
A fair six-sided die is rolled 5 times. What is the probability of obtaining exactly two sixes?
At which point(s) does the curve $y = x^{3} - 3 x$ have a horizontal tangent line?
Find the derivative of $f (x) = 5^{2 x}$ .
Evaluate the definite integral $\int_{0}^{1} \frac{1}{1 + x ^{2}} d x$ .
Evaluate $\int_{0}^{π /6} cos (3 x) d x$ .
For the function $f (x) = x^{4} - 4 x^{3}$ , find the point of inflection where concavity changes.
Find the value of $c \in (1, 4)$ satisfying the Mean Value Theorem for $f (x) = x^{2} - 2 x$ .
Evaluate the sequence limit $lim_{n \to \infty} (1 + \frac{2}{n})^{n}$ .
Find the average value of $f (x) = 3 x^{2} - 2 x$ on the interval $[0, 2]$ .
Determine the derivative of $f (x) = arctan (3 x)$ .
Set up the integral for the arc length of $y = x^{2}$ from $x = 0$ to $x = 1$ .
Find the volume of the solid generated by revolving $y = x$ about the $x$ -axis from $x = 0$ to $x = 4$ .
Compute the derivative of $f (x) = sin^{2} (x)$ .
Evaluate $\int_{1}^{e^{2}} \frac{1}{x} d x$ .
Find the second-degree Taylor polynomial $P_{2} (x)$ for $f (x) = e^{x}$ centered at $x = 0$ .
Evaluate $lim_{x \to 0} \frac{t a n ( 5 x )}{x}$ .

Differential and Integral Calculus, Differential Equations, and Probability

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