Panhellenic Examinations (Greece) · Mathematics

Differential Calculus and Curve Sketching

14 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 14 questions

What is the derivative of $f (x) = ln (x^{2} + 4)$ ?
What is the equation of the tangent line to $f (x) = x^{3} - 3 x + 2$ at the point where $x = 2$ ?
At which critical point does $f (x) = x^{3} - 12 x$ attain a local minimum?
On which interval is the function $f (x) = x e^{- x}$ strictly increasing?
Rolle's theorem applied to $f (x) = x^{2} - 4 x + 3$ on the interval $[1, 3]$ guarantees a point $c$ where $f^{'} (c) = 0$ . What is $c$ ?
Where is the inflection point of the cubic polynomial $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ located?
What is the derivative of $f (x) = tan (3 x)$ ?
What is the Mean Value Theorem guarantee for $f (x) = x^{2}$ on $[0, 4]$ ? Find $c \in (0, 4)$ where $f^{'} (c) = \frac{f ( 4 ) - f ( 0 )}{4 - 0}$ .
If $f^{'} (x) > 0$ on $(a, b)$ and $f^{''} (x) < 0$ on $(a, b)$ , how does the graph of $f$ behave?
What is the derivative of $f (x) = x^{x}$ for $x > 0$ ?
A particle moves along a line with position $s (t) = t^{3} - 6 t^{2} + 9 t$ . At what times is the particle at rest?
What is the maximum area of a rectangle with perimeter $20$ meters?
Fermat's theorem on stationary points states that if $f$ has a local extremum at an interior point $c$ and is differentiable at $c$ , then:
What is the derivative of $f (x) = arcsin (x)$ for $x \in (- 1, 1)$ ?

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