Panhellenic Examinations (Greece) · Μαθηματικά Προσανατολισμού

Advanced Differentiation and Theoretical Calculus

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

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All 14 questions

If $x^{2} + y^{2} = 25$ , what is the second derivative $\frac{d ^{2} y}{d x ^{2}}$ at the point $(3, 4)$ ?
Let $f$ be twice differentiable on $R$ with $f^{''} (x) > 0$ everywhere. Which statement about the tangent lines is true?
What is the derivative of $f (x) = ln (sec x + tan x)$ for $x \in (- π /2, π /2)$ ?
On which interval is the function $f (x) = x - ln x$ strictly increasing?
If $f$ is differentiable on $[0, 2]$ with $f (0) = 1$ and $f^{'} (x) \leq 3$ for all $x \in (0, 2)$ , what is the maximum possible value of $f (2)$ ?
How many inflection points does the polynomial $f (x) = x^{4} - 6 x^{2} + 5$ possess?
What is the limit $lim_{x \to 0} \frac{x - s i n x}{x ^{3}}$ ?
What is the acute angle between the curves $y = x^{2}$ and $y = x$ at their intersection point $(1, 1)$ ?
If $f : R \to R$ is differentiable and satisfies $f (x) + e^{f (x)} = 2 x + 1$ , what is $f^{'} (0)$ given $f (0) = 0$ ?
What is the derivative of $f (x) = arctan (e^{x})$ ?
Find the coordinates of the local maximum of $f (x) = \frac{l n x}{x}$ for $x > 0$ .
What is the slant asymptote of $f (x) = x^{2} + 4 x$ as $x \to + \infty$ ?
Let $f : [a, b] \to R$ be continuous and differentiable on $(a, b)$ . Darboux's theorem on derivatives states that:
What is the derivative of $f (x) = x^{s i n x}$ for $x > 0$ ?

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