Riigieksamid (Estonian State Exams) · Mathematics

Differential and Integral Calculus

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

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All 11 questions

Differentiate $f (x) = x^{2} e^{3 x}$ with respect to $x$ .
What is the equation of the normal line to the curve $y = x^{3} - 3 x + 2$ at the point where $x = 2$ ?
Evaluate the definite integral $\int_{0}^{1} \frac{1}{1 + x ^{2}} d x$ .
At what value of $x$ does the function $f (x) = \frac{l n x}{x}$ achieve its absolute maximum on $(0, \infty)$ ?
Calculate the area of the region bounded by $y = 6 x - x^{2}$ and the $x$ -axis.
Find the antiderivative $\int x cos (x) d x$ using integration by parts.
What is the limit $lim_{x \to 0} \frac{e ^{2 x} - 1}{s i n ( 3 x )}$ ?
Find all inflection points of the function $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ .
Evaluate the improper integral $\int_{1}^{\infty} \frac{1}{x ^{3}} d x$ .
What is the volume of the solid generated by revolving the region bounded by $y = x$ , $y = 0$ , and $x = 4$ about the $x$ -axis?
Find the derivative of $g (x) = arctan (2 x)$ .

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