Chapter 17: Exam
- Page ID
- 212862
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
Part I (Practice, 4 hours)
Let \(U=\left\{\frac{1}{3^{n^2}}\right\}_{n\in \mathbb{N}_0}\) and \(\mathbb{T}=(1-U)\cup (1+U)\cup \{2\}\cup (2+U)\).
1. Prove that \(\mathbb{T}\) is a time scale.
2. Find the forward jump operator \(\sigma\).
3. Find the backward jump operator \(\rho\).
4. Classify each points of \(\mathbb{T}\).
5. Find the forward graininess function \(\mu\).
6. Find the backward graininess function \(\nu\).
7. Find \(\sigma^2(\rho(t))\) for \(t\in (1-U)\).
8. Find \(\mu(t)-\nu(t)\rho^3(\sigma^2(t))\) for \(t\in (2+U)\).
9. Simplify
\[A(t)= \left(\frac{5\sigma(t)-2(\sigma(t))^2}{(\sigma(t)-2)^2}+\sigma(t)\right): \left(\sigma(t)-\frac{3\sigma(t)-8}{(\sigma(t)-2)^2}-2\right), \quad t\in \mathbb{T},\quad \sigma(t)\ne 2.\notag\]
10. Check if the following inequality is true
\[\sigma^4(t)\rho^2(t)-\rho^3(t)+(\mu(t))^2-t\leq \sigma(t),\quad t\in (1-U).\notag\]
Let \(\mathbb{T}=2^{\mathbb{N}_0}\cup \left(\frac{1}{3}\right)^{\mathbb{N}_0}\cup 4\mathbb{Z}\) and
\[ f(t)= \frac{a+bt+ct^2}{d+kt},\quad t\in \mathbb{T},\notag\]
where \(a, b, c, d, k\) are real parameters.
1. For \(a=0, b=1, c=3, d=-1, k=3\), find
\[ f^\rho(t)-f^{\sigma^2\rho^3}(t),\quad t\in \left(\frac{1}{3}\right)^{\mathbb{N}_0}.\notag\]
2. For \(a=-1, b=0, c=1, d=3, k=2\), find
\[ f^{\sigma^2}(t),\quad t\in 2^{\mathbb{N}_0}.\notag\]
3. For \(a=2, b=3, c=-4, d=1, k=-1\), check if the function is rd-continuous.
4. For \(a=-5, b=-1, c=1, d=0, k=4\), check if the function is ld-continuous.
5. For \(a=-\frac{1}{3}, b=2, c=0, d=-1, k=5\), check if the function is continuous.
6. For \(a=2, b=-1, c=2, d=-3, k=1\), check if the function is rd-regulated.
7. For \(a=23, b=1, c=0, d=5, k=1\), check if the function is predifferentiable.
8. For \(a=-10, b=1, c=2, d=3, k=2\), check if the function is regressive.
9. For \(a=15, b=0, c=4, d=-1, k=1\), check if the function is ld-regulated.
10. For \(a=-2, b=1, c=3, d=10, k=1\), find
\[(f(t)\oplus_\mu \sigma(t))\ominus_\nu \rho(t),\quad t\in 4\mathbb{Z}.\notag\]
Let \(\mathbb{T}=\left(\frac{1}{2}\right)^{\mathbb{N}_0}\cup 4^{\mathbb{N}_0}\) and
\[\begin{aligned}f(t)=& 2-3t+4t^2-t^3,\\g(t)=& \frac{1+2t}{2+7t},\quad t\in \mathbb{T}.\end{aligned}\notag\]
1. Find \(f^\Delta(t)\) for \(t\in \left(\frac{1}{2}\right)^{\mathbb{N}_0}\).
2. Find \(f^{\nabla^2}(t)\) for \(t\in 4^{\mathbb{N}_0}\).
3. Find \(\left(\frac{f}{g}\right)^\Delta(t)\) for \(t\in \left(\frac{1}{2}\right)^{\mathbb{N}_0}\).
4. Find \((fg)^\nabla(t)\) for \(t\in 4^{\mathbb{N}_0}\).
5. Find the region of delta differentiation of the function f in \(\left(\frac{1}{2}\right)^{\mathbb{N}_0}\).
6. Find the region of nabla differentiation of the function \(g\) in \(4^{\mathbb{N}_0}\).
7. Find the regions of increasing of the function \(f\) over the whole time scale \(\mathbb{T}\).
8. Find the regions of decreasing of the function \(f\) in \(4^{\mathbb{N}_0}\).
9. Investigate for convexity the function \(f-g\) over the whole time scale \(\mathbb{T}\).
10. Investigate for concavity the function \(fg\) over the whole rime scale \(\mathbb{T}\) and investigate it for local extremum points.
Let \(\mathbb{T}=(-5\mathbb{N}_0)\cup 4^{\mathbb{N}_0}\) and
\[f(t)=\begin{cases} 3t^2+15t+25\quad \mbox{if}\quad t\in (-5\mathbb{N})\\ 1\quad \mbox{if}\quad t=0\\ 21t^2\quad \mbox{if}\quad t\in 4^{\mathbb{N}_0}\end{cases}\notag\]
and
\[ g(t)=at^2+bt+c,\quad t\in \mathbb{T},\notag\]
where \(a, b, c\) are real parameters.
1. Using the definition for delta integral, prove that
\[\int f(t)\Delta t=t^3+d,\quad t\in \mathbb{T},\notag\]
where \(d\) is a constant.
2. For \(a=-1, b=2, c=3\), find
\[\int\limits_{-10}^{64}g(t)\Delta t.\notag\]
3. Find
\[\int\limits_{-625}^0 f(t)\nabla t.\notag\]
4. Check the rule for integration by parts for the functions \(f\) and \(g\) in \(4^{\mathbb{N}_0}\) for \(a=1, b=0, c=-3\).
5. For \(a=1, b=-2, c=1\), investigate for convergence the integral
\[\int\limits_0^{16}\frac{1}{g(t)}\Delta t.\notag\]
6. For \(a=2, b=3, c=-1\) write the Taylor formulae to the fourt term for the function \(g\).
7. For \(a=2, b=-3, c=1\), find
\[\int\limits_{-5}^{16}g(t)h_2(t, -5)\Delta t,\notag\]
where \(h_2\) is the generalized time scales monomial.
8. For \(a=1, b=3, c=2\), find
\[\int\limits_1^{64}(e_2(t, 1)-\sinh_3(t, 1)-3g(t))\Delta t. \notag\]
9. For \(a=0, b=0, c=2\), find
\[\int_{-10}^1 (g(t)e_1(t, -10)-h_2(t, -10)) \Delta t.\notag\]
10. For \(a=3, b=0, c=10\), find
\[\int_1^{128}\left((g(t))^2-f(t)\right)\Delta t.\notag\]
Part II (Theory, 4 hours)
Formulate and prove the dual induction principle.
Give a definition for first order nabla derivative and prove that the definition is correct.
Prove the rule for delta differentiation of product of two functions.
Give an example for a forward jump operator that is not continuous.
Formulate and prove the Rolle theorem in the nabla case.
Give a definition for a completely delta differentiable function and formulate and prove a criterion for completely delta differentiablity of a function.
Formulate and prove the first mean value theorem.
Give a definition for Riemann delta integrable function.
Formulate and prove the Cauchy criterion for Riemann nabla integrability of a function.
Give definitions for the monomials \(h_k\) and \(g_k\) and deduct a relation between them.