VWO / HAVO Centraal Examen (Netherlands) · Mathematics

Calculus and Optimization

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

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All 11 questions

What is the derivative of $f (x) = x^{3} ln (x)$ with respect to $x$ for $x > 0$ ?
What is the equation of the tangent line to the curve $y = x^{3} - 3 x + 2$ at the point where $x = 2$ ?
Evaluate the definite integral $\int_{1}^{e} \frac{1}{x} d x$ .
What is the total area bounded by the parabola $y = x^{2}$ and the line $y = 2 x$ ?
A rectangular garden is to be fenced using 40 metres of fencing along three sides, with an existing brick wall forming the fourth side. What is the maximum possible area of the garden?
Evaluate the improper integral $\int_{0}^{\infty} e^{- 2 x} d x$ .
What is the second derivative of the trigonometric function $f (x) = cos (2 x)$ ?
What is the volume of the solid generated by revolving the region under $y = x^{2}$ from $x = 0$ to $x = 2$ about the x-axis?
Using L'Hôpital's rule, evaluate the indeterminate limit $lim_{x \to 0} \frac{e ^{3 x} - 1}{s i n ( 2 x )}$ .
What is the antiderivative of $f (x) = \frac{1}{1 + x ^{2}}$ with respect to $x$ ?
For the cubic function $f (x) = 2 x^{3} - 9 x^{2} + 12 x + 1$ , what are the coordinates of the local minimum?

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