EmSAT Achieve (UAE) · Mathematics

Differential Calculus and Probability Distributions

28 practice questions, each with the answer and an explanation.

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What is the derivative of $f (x) = 4 x^{3} - 5 x^{2} + 7 x - 9$ with respect to $x$ ?
What is the derivative of $g (x) = e^{3 x^{2} - 2}$ with respect to $x$ ?
Evaluate $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$ .
At what point on the curve $y = x^{2} - 6 x + 8$ is the tangent line horizontal?
If a fair standard 6-sided die is rolled twice, what is the probability that the sum of the two numbers is equal to $7$ ?
In a binomial distribution with $n = 100$ trials and probability of success $p = 0.36$ , what is the standard deviation $σ$ ?
What is the equation of the tangent line to $y = x^{3} - 2 x + 1$ at the point where $x = 1$ ?
Evaluate $\int (6 x^{2} - 4 x + 3) d x$ .
Evaluate the definite integral $\int_{1}^{3} (3 x^{2} - 2 x) d x$ .
A random variable $X$ follows a normal distribution with mean $μ = 50$ and standard deviation $σ = 8$ . What is the $z$ -score corresponding to $X = 66$ ?
What is the critical point of $f (x) = x^{3} - 12 x + 7$ on the interval $(0, \infty)$ ?
If events $A$ and $B$ are independent with $P (A) = 0.4$ and $P (B) = 0.5$ , what is $P (A \cup B)$ ?
What is the inflection point of the cubic curve $y = x^{3} - 6 x^{2} + 9 x - 2$ ?
Evaluate $lim_{x \to 0} \frac{s i n ( 4 x )}{x}$ .
What is the area of the region bounded by $y = x^{2}$ and $y = 2 x$ ?
If a continuous random variable $X$ has probability density function $f (x) = k x$ for $0 \leq x \leq 2$ (and $0$ elsewhere), what is the value of $k$ ?
What is the derivative of $y = ln (x^{2} + 4)$ with respect to $x$ ?
A bag contains 5 red balls and 3 blue balls. If two balls are drawn at random without replacement, what is the probability that both are red?
What is the second derivative $\frac{d ^{2} y}{d x ^{2}}$ of $y = cos (3 x)$ ?
Evaluate $lim_{x \to \infty} \frac{3 x ^{2} + 5 x}{7 x ^{2} - 2}$ .
What is the rate of change of the volume $V$ of a sphere with respect to its radius $r$ when $r = 3$ (given $V = \frac{4}{3} π r^{3}$ )?
What is the expected value $E (X)$ of a discrete random variable $X$ with the probability distribution:
$X = 1$ with $P = 0.2$
$X = 2$ with $P = 0.5$
$X = 3$ with $P = 0.3$ ?
Evaluate $\int e^{2 x} d x$ .
What is the average value of the function $f (x) = 2 x + 1$ on the interval $[0, 4]$ ?
If random variable $X \sim N (50, 16)$ , what is $P (X \leq 58)$ ? (Use standard normal $Φ (2) \approx 0.9772$ )
What is the cross product $u \times v$ of $u = ⟨ 1, 0, 2 ⟩$ and $v = ⟨ 0, 3, 1 ⟩$ ?
Evaluate $\int x e^{2 x} d x$ .
What is the remainder when $P (x) = 2 x^{4} - 3 x^{3} + 5 x - 7$ is divided by $(x - 2)$ ?

Differential Calculus and Probability Distributions

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