Panhellenic Examinations (Greece) · Mathematics

Calculus: Limits, Derivatives and Integrals

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

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 36 questions

Which set of conditions is required by Bolzano's theorem to guarantee a root of $f$ in $(a, b)$ ?
What is the derivative of $f (x) = x ln x$ for $x > 0$ ?
What is $lim_{x \to + \infty} \frac{2 x + 1}{x - 3}$ ?
Evaluate $\int_{0}^{2} 3 x^{2} d x$ .
What is the derivative of $f (x) = sin 2 x$ ?
For $f (x) = x^{2} - 4 x + 3$ on $[1, 3]$ , Rolle's theorem applies. Which $ξ \in (1, 3)$ satisfies $f^{'} (ξ) = 0$ ?
For $f (x) = x^{2}$ on $[1, 3]$ , which $ξ \in (1, 3)$ does the Mean Value Theorem give?
Fermat's theorem says that if $f$ is differentiable at an interior point $x_{0}$ where it has a local extremum, then $f^{'} (x_{0}) = 0$ . Which example shows that the converse is false?
On which interval is $f (x) = x^{3} - 3 x$ strictly decreasing?
The function $f (x) = {\frac{s i n x}{x} a x \neq = 0 x = 0$ is continuous at $0$ . What is $a$ ?
What is the derivative of $f (x) = e^{x^{2}}$ ?
Evaluate $\int_{1}^{e} \frac{1}{x} d x$ .
Evaluate $\int_{0}^{π} sin x d x$ .
What is the area under $y = x$ between $x = 0$ and $x = 4$ ?
Let $F (x) = \int_{1}^{x} t^{2} d t$ . What is $F^{'} (3)$ ?
What is $lim_{x \to 0} \frac{e ^{x} - 1}{x}$ ?
What is the equation of the tangent line to $f (x) = e^{x}$ at $x = 0$ ?
What is the oblique asymptote of $f (x) = \frac{x ^{2} + 1}{x}$ as $x \to + \infty$ ?
What is the derivative of $f (x) = ln (sin x)$ on $(0, π)$ ?
What is $lim_{x \to 0} \frac{1 - c o s x}{x ^{2}}$ ?
A particle moves with position $s (t) = t^{3} - 6 t^{2} + 9 t$ for $t \geq 0$ . At which times is its velocity zero?
A rectangle has area $100$ . What is its smallest possible perimeter?
What is $\int ln x d x$ for $x > 0$ ?
If $f^{'} (x) = 0$ for every $x$ in an interval, what can we conclude?
What is the domain of $f (x) = ln (x^{2} - 4)$ ?
What is the derivative of $f (x) = \frac{x}{x + 1}$ ?
Does $f (x) = x^{4}$ have an inflection point at $x = 0$ ?
How many real roots does $f (x) = x^{3} + x - 1$ have?
A box with no top is made from a $12 \times 12$ sheet by cutting a square of side $x$ from each corner and folding up the sides. What is the largest possible volume?
Evaluate $\int_{0}^{1} x e^{x^{2}} d x$ .
For $g (x) = e^{x} + x - 3$ , which argument proves that $e^{x} = 3 - x$ has exactly one real solution?
What is the maximum value of $f (x) = ln x - x + 1$ for $x > 0$ ?
Evaluate $\int_{0}^{2} ∣ x - 1∣ d x$ .
What is $lim_{x \to 0^{+}} x ln x$ ?
Let $f (x) = e^{x} + x$ and let $f^{- 1}$ be its inverse. What is $(f^{- 1})^{'} (1)$ ?
What is the area enclosed by $y = x$ and $y = x^{3}$ for $0 \leq x \leq 1$ ?

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