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\begin{document}



\section*{Derivatives}

\begin{zad}
Show that the function $y=c_1e^{2x}+c_2xe^{2x}+e^x$ satisfies the differential equation $y''-4y'+4y=e^x$.
\end{zad}

\begin{zad}
Show that the function $y=e^{-x}\sin x$ satisfies the differential equation $y''+2y'+2y=0$.
\end{zad}

\begin{zad}
Calculate the derivative of the function:
$$
y=-\frac{\cos x}{\sin^2 x}+\ln\tg\frac{x}{2}.
$$
\end{zad}

\section*{Equation of the Tangent to a Graph}

\begin{zad}
Find the equation of the tangent and normal to the parabola $y=4-x^2$ at the point where $x=-1$.
\end{zad}

\begin{zad}
Find the equation of the tangent to the graph $y=\arcsin x$ at the point $x=\frac{1}{2}$.
\end{zad}

\begin{zad}
Find the intersection of the tangents to the curve $y=\frac{1}{1+x}$ at the points $x=-3$ and $x=-\frac{1}{2}$.
\end{zad}

\begin{zad}
Prove that the length of the tangent segment between the coordinate axes for an astroid is constant! The astroid is given by the equation $x^{2/3}+y^{2/3}=\alpha^{2/3}$, where $\alpha$ is a positive number.
\end{zad}

\begin{zad}
Find the normal to the graph of the real function $f(x) = x^2 + 3x - 1$ that intersects it at the point with the abscissa $x = 0$.
\end{zad}

\section*{Linear Approximation}

\begin{zad}
Using the differential (linear approximation), approximate $\sqrt{1.05}$.
\end{zad}

\begin{zad}
Using the differential, approximate the value of $\ln(0.95)$.
\end{zad}

\begin{zad}
Using the differential (linear approximation), approximate $\sqrt[3]{8.05}$.
\end{zad}

\begin{zad}
Using the differential (linear approximation), approximate $\cos\left(\frac{2\pi}{9}\right)$.
\end{zad}

\begin{zad}
Using the differential (linear approximation), approximate $\sqrt{21}$. Solve the problem by considering the function $f(x)=\sqrt{10x-x^2}$ at the point $x_0=2$.
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}





\newpage
\section*{Taylor Polynomial}

\begin{zad}
Derive the Taylor polynomial $T_2(x)$ (order 2) for the function $e^x$ around $0$. Using the Taylor polynomial $T_2(x)$, approximate $e^{0.12}$ and estimate the error $R_2(x)$. %%(Recall $f(x)=T_2(x)+R_2(x)$, where $f(x)$ is the exact value, $T_2(x)$ is the second-order approximation, and $R_2(x)$ is the corresponding error).
\end{zad}

\begin{zad}
Derive the Taylor polynomial $T_2(x)$ (order 2) for the function $\ln x$ around $1$. Using the Taylor polynomial $T_2(x)$, approximate $\ln(0.95)$.
\end{zad}

\begin{zad}
Derive the Taylor polynomial $T_2(x)$ (order 2) for the function $f(x)=\frac{1}{\sqrt{4+x}}$ around $0$. Using the Taylor polynomial $T_2(x)$, approximate $(4.13)^{-\frac{1}{2}}$ and estimate the error $R_2(x)$.
\end{zad}

\section*{Finding Extrema}

\begin{zad}
Find the global maximum and global minimum of the function $f(x)=10x\,(2-\ln x)$ on the interval $[1, e^2]$.
\end{zad}

\begin{zad}
Determine and classify the local extrema of the function
$$
g(x)=\frac{(2x-1)^3}{(x+2)^2}.
$$
\end{zad}

\begin{zad}
The function $f(x)=\frac{x}{x^2+ax+b}$ is given. Find constants $a$ and $b$ such that $T(2, \frac{1}{7})$ is a local extremum.
\end{zad}

\begin{zad}
Determine the domain, sign, and local extrema of the function
$$
y=\ln\frac{x}{x^2-1}.
$$
\end{zad}

\begin{zad}
Let $p$ be a line passing through the point $P(6, 3)$ in the $xy$-plane. Consider a right triangle formed by the line $p$, the positive $x$-axis, and the positive $y$-axis. Find the minimum possible area of the given right triangle.

{\it (Hint: Notice that the solution is not the line $p$ passing through $(0, 0)$ because in that case, the line does not intersect the positive $x$-axis or the positive $y$-axis).}
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}

\newpage
\section*{L'Hospital's Rule}

\begin{zad}
Using L'Hospital's rule, calculate the limit
$$
\lim_{x\to 0}\frac{1-\cos ax}{1-\cos bx}.
$$
\end{zad}

\begin{zad}
Calculate
$$
\lim_{x\downarrow 1}\left(
\frac{1}{x-1}-\frac{1}{\ln x}
\right).
$$
\end{zad}

\begin{zad}
Calculate
$$
\lim_{x\to 0}\frac{x^2+e^{x^3}-1}{\ln(1+x)-\sin x}.
$$
\end{zad}

\begin{zad}
Calculate
$$
\lim_{n\to\infty} 4^n\cdot (1-\cos\frac{\theta}{2^n}).
$$
\end{zad}

\section*{Convexity and Concavity of Functions}

\begin{zad}
Determine the domain, sign, and intervals of convexity and concavity for the function $y=xe^{\frac{1}{x}}$.
\end{zad}

\begin{zad}
The function $f(x)=x\,e^{\frac{1}{x-2}}$ is given. Determine its domain, sign, and local extrema, and find intervals of convexity and concavity. Explain the geometric meaning of the limit
$$
\lim_{x \downarrow 2} f(x),
$$
and calculate it.
\end{zad}

\begin{zad}
Find the intervals where the function
$$
f(x)=\frac{x^2+1}{(x-1)^2}
$$
is strictly increasing, and find the intervals of convexity and concavity. Explain the geometric meaning of the following limits:
$$
\lim_{x\to\infty} f(x),\quad
\lim_{x\to-\infty} f(x),\quad
\lim_{x\uparrow 1} f(x),\quad
\lim_{x\downarrow 1} f(x),
$$
and calculate them.
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}






\newpage
\section*{Graph Sketching of Functions}

\begin{zad}
Carefully sketch the graph of the function 
$$
f(x)=\frac{x}{x-3}
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function 
$$
f(x)=\frac{3x}{1+x^3}
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function
$$
f(x)=\frac{x-2}{x^2-3}
$$
(Determine the domain, zeros, sign of the function, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points, if they exist).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function 
$$
f(x)=\frac{x^2-6x+3}{x-3}
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function 
$$
y=\frac{1}{x^2}\ln x
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function 
$$
y=\frac{1}{x-3}\ln x
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function 
$$
f(x)=\frac{e^x-e^{-x}}{e^x}
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\begin{zad}
Carefully sketch the graph of the function 
$$
y=x^3e^{-\frac{x^2}{6}}
$$
(Determine the domain, zeros, limits at the boundaries of the domain, asymptotes, intervals of increase and decrease, local extrema, intervals of convexity and concavity, and inflection points).
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}





\newpage
\section*{Integration by Substitution}

\begin{zad}
Compute the following integral:
$$
\int \frac{\sin x \, dx }{\sqrt{1+2\cos x}}.
$$
\end{zad}

\begin{zad}
Compute:
$$
\int \tan\varphi \, d\varphi.
$$
\end{zad}

\begin{zad}
Compute:
$$
\int \sin x \, \cos^3\!x \, dx,
$$
and verify the result by differentiation.
\end{zad}

\section*{Integration by Parts}

\begin{zad}
Compute the following integral:
$$
\int x\,(\ln x)^2\, dx.
$$
\end{zad}

\begin{zad}
Compute the following integral:
$$
\int (3x+4)\,\sin(2x) \, dx.
$$
\end{zad}

\section*{Integration of Rational Functions}

\begin{zad}
Determine:
$$
\int \frac{dx}{x^3+x}.
$$
\end{zad}

\begin{zad}
Compute:
$$
\int_{0}^{1} \frac{x^4}{x^2+1} \, dx.
$$
\end{zad}

\begin{zad}
Determine:
$$
\int \frac{x^2-x+1}{x^3+x} \, dx.
$$
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}






\newpage
\section*{Definite Integral Concept}

\begin{zad}
Calculate the area of the region bounded by the curve $y = x^2 - 5x + 6$ and the two coordinate axes.
\end{zad}

\begin{zad}
Compute:
$$
\int_{-1}^2 |x^2-1|\,dx
$$
and explain its geometric meaning.
\end{zad}

\begin{zad}
Compute:
$$
\int_{-2}^1 |x^2-x| \, dx
$$
and explain its geometric meaning.
\end{zad}

\begin{zad}
Compute:
$$
\int_{-1}^2 |x^3-x|\,dx
$$
and explain its geometric meaning.
\end{zad}

\begin{zad}
Compute the following definite integral:
$$
\int_0^2 \frac{e^x}{e^{2x}+1} \, dx.
$$
\end{zad}

\section*{Integral as a Function of the Upper Limit}

\begin{zad}
Find the indefinite integral of the function $f(t)=\cot t$ such that its value at $t=\frac{\pi}{4}$ is $0$.
\end{zad}

\section*{Areas of Regions}

\begin{zad}
The region $L$ is bounded by the lines $x=1$, $x=e^2$, the function $y=\frac{\ln x}{x^2}$, and the $x$-axis. Compute the area of region $L$.
\end{zad}

\begin{zad}
Calculate the area of the closed region enclosed by the curves $y = x^2 + 4x$ and $y = x + 4$.
\end{zad}

\begin{zad}
Find the area of the region bounded by the parabola $4y = 8x - x^2$ and the line $4y = x + 6$.
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}





\newpage
\section*{Volumes of Solids of Revolution}

\begin{zad}
Calculate the volume of the solid generated by rotating the graph of the function $f(x) = x+\sin x$ about the $x$-axis over the interval $[0,\pi]$.
\end{zad}

\begin{zad}
The region $L$ is bounded by the lines $x=0$, $y=5$, and the parabola $y=x^2+1$. Compute the volume of the solid obtained by rotating region $L$ about the $y$-axis.
\end{zad}

\begin{zad}
Find the volume of the solid of revolution generated by rotating the region bounded by the curves $xy = 4$, $x = 4$, $x = 1$, and $y = 0$ about the $x$-axis.
\end{zad}

\begin{zad}
Compute the volume of the solid obtained by rotating the region enclosed between the graphs of $f(x) = x^2 + 2x + 2$ and $g(x) = x+4$ about the $x$-axis.
\end{zad}

\section*{Arc Length of a Curve}

\begin{zad}
Find the length of the graph of the function $y = \ln\left(\frac{1}{\cos x}\right)$ over the interval $[0,\frac{\pi}{4}]$.
\end{zad}

\begin{zad}
Compute the arc length of the curve $f(x)=\frac{e^x+e^{-x}}{2}$ over the interval $[0,1]$.
\end{zad}

\begin{zad}
Find the length of the curve defined by $y^2=(x-1)^3$ between the points $A(2,-1)$ and $B(5,-8)$.
\end{zad}

\section*{Surface Area of Solids of Revolution}

\begin{zad}
Let $C$ be the part of the graph of the function $y = x^3$ lying between the lines $x=-\frac{2}{3}$ and $x=\frac{2}{3}$. Compute the surface area of the solid generated by rotating the curve $C$ about the $x$-axis.
\end{zad}

\begin{zad}
Calculate the surface area of the solid generated by rotating the graph of $y = \sin x$ about the $x$-axis between two consecutive zeros.
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}

\newpage
\section*{Improper Integrals}

\begin{zad}
Determine the domain of the function $f(x)=\frac{1}{\sqrt[3]{(x-1)^2}}$. Then compute the following integral and explain its geometric meaning:
$$
\int_{-1}^2 \frac{\dx}{\sqrt[3]{(x-1)^2}}.
$$
\end{zad}

\section*{Sequence of Functions and Uniform Convergence}

\begin{zad}
The sequence of functions $(f_n)_{n\in{\mathbb N}}$ is defined by:
$$
f_n:[0,1]\rightarrow\mathbb{R}, \qquad f_n(x)=2x+\frac{x}{n}.
$$
\begin{enumerate}[label=\rm(\alph*)]
\item Determine the limit function $f(x)=\lim_{n\to\infty} f_n(x)$.
\item Is the limit function $f(x)$ continuous on $[0,1]$? Provide a detailed explanation.
\item Verify whether $(\lim_{n\to\infty} f_n(x))'=\lim_{n\to\infty} f'_n(x)$ holds for all $x\in[0,1]$.
\item Check if $\int_0^1 \lim_{n\to\infty} f_n(x)\dx=\lim_{n\to\infty} \int_0^1 f_n(x)\dx$.
\end{enumerate}
\end{zad}

\begin{zad}
The sequence of functions $f_n : [0,\infty)\to\mathbb{R}$ is given by $f_n(x) = \frac{nx}{1+nx}$. Determine the limit function and establish whether the sequence converges uniformly on $[a,\infty)$, where $a>0$. Does the sequence converge uniformly on $[0,\infty)$?
\end{zad}

\begin{zad}
The sequence of functions $(f_n)_{n\in{\mathbb N}}$ is defined by:
$$
f_n:[0,1]\to\mathbb{R}, \qquad f_n(x)=\frac{1}{2}x^n.
$$
\begin{enumerate}[label=\rm(\alph*)]
\item To which function does the sequence $(f_n)_{n\in{\mathbb N}}$ converge on $[0,1]$?
\item Is the convergence in (a) uniform?
\end{enumerate}
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}





\newpage
\section*{Generalized (Improper) Integrals}

\begin{zad}
Evaluate the following integral if it converges:
$$
\int_{-\infty}^{+\infty} \frac{|x|}{1+x}\dx.
$$
\end{zad}

\begin{zad}
Check whether the improper integral $\int_{-\infty}^0 \frac{e^{1/x}}{x^2}\dx$ converges. If it does, compute its value.
\end{zad}

\section*{Integral Test}

\begin{zad}
Using the integral test, determine whether the series
$$
\sum_{n=1}^{+\infty} \frac{2n}{n^2+1}
$$
converges.
\end{zad}

\begin{zad}
Using the integral test, determine whether the series
$$
\sum_{n=2}^\infty \frac{1}{n\, \ln n}
$$
converges.
\end{zad}

\begin{zad}
Using the integral test, determine whether the series
$$
\sum_{n=1}^\infty \frac{\ln n}{n}
$$
converges.
\end{zad}

\begin{zad}
Using the integral test, determine whether the series
$$
\sum_{n=1}^\infty \arcsin\frac{1}{\sqrt{n}}
$$
converges.
\end{zad}

\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}



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