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In this question you will test your knowledge of the chain rule. You will compute the rate of change of height $z=f(x,y)$ when moving along a space curve ${\\bf r}(t) = (x(t),y(t),z(x(t),y(t))), t \\in [a,b].$

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Let $f(x,y)=x^{\\var{n1}}y^{\\var{m1}}+x^{\\var{n2}}\\exp(\\var{-m2} y)$. Let $x(t)=t^{\\var{n3}}$ and $y=\\var{a} t^{\\var{m3}}$. If $z=f(x,y)$, compute ${dz \\over dt}$ when $t = \\var{s}$. Give your answers to 2 decimal places.

When $t=\\var{s}$, ${\\partial f \\over \\partial x}=$[[0]].

When $t=\\var{s}$, ${\\partial f \\over \\partial y}=$[[1]].

When $t=\\var{s}$, ${d x \\over dt}=$[[2]].

When $t=\\var{s}$, ${d y \\over dt}=$[[3]].

Thus, when $t=\\var{s}$, ${dz \\over dt}=$[[4]].

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In this question you will calculate the gradient vector for polynomial surfaces of the form $z=ax^{n1} y^{m1} + b x^{n2} y^{m2}$. You will then use this information to calculate the rate of change of these functions in the direction of a vector ${\\bf n}$. You will need to create a unit vector in the direction of this vector ${\\bf n}$. Put in your answers to 2 decimal places.

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Let $z=f(x,y)$, where $f(x,y)=\\var{a}x^{\\var{n1}} y^{\\var{m1}}+\\var{b}x^{\\var{n2}} y^{\\var{m2}}$. Calculate ${\\partial z \\over \\partial {\\bf n}}$, at $P=(\\var{s},\\var{t})$, where ${\\bf n}=(\\var{c},\\var{d})$.

The gradient vector at $P$ is ([[0]],[[1]]).

A unit vector in the direction of ${\\bf n}$ is ([[2]],[[3]]).

The rate of change of $z$ in the direction of ${\\bf n}$ at $P$ is [[4]].

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In this question you will find the equation of the tangent plane for at a point $P=(c,d)$ on a surface $z=ax^{n1}y^{m1}+bax^{n2}y^{m2}$. The equation of the plane is in the form $z=ex+fy+g$. You will find the numbers $e$, $f$ and $g$ to 2 decimal places.

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Let $z=f(x,y)$, where $f(x,y)=\\var{a}x^{\\var{n1}}y^{\\var{m1}}+\\var{b}x^{\\var{n2}}y^{\\var{m2}}$. Find the tangent plane to the surface $z=f(x,y)$ at $P=(\\var{c},\\var{d})$.

The equation of the tangent plane is:

$z=$[[0]]$x$ + [[1]]$y$ + [[2]].

This question will test your understanding of upper bound, lower bound, infimum (greatest lower bound), supremum (least upper bound), maximum, and minimum.

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Let $(\\var{a},\\var{b})$ be an open interval. Which of the following statements are true?

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Let $[\\var{a},\\var{b}]$ be a closed interval. Which of the following statements is true?

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Write the following inequality in the form $\\pm 1 x<A$, where $A$ is a number you have to find.

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Solve the linear inequality $\\var{a}x+\\var{b}<\\var{c}x+\\var{d}$. In the first gap put $\\pm 1$ to make the inequality correct.

[[1]]$x$<[[0]]

Find the ranges of $x$ for which $\\var{qa}x^2+\\var{qb}x+\\var{qc}>0$.

$x<$ [[1]]

$x>$ [[0]]

Which of the statements below is one of the two triangle inequalities? If you are in any doubt, I suggest that you put actual numbers (including positive, negative numbers and 0) into the inequality to check whether or not it is true. Remember that even if it is true for the numbers you select it does not mean it is true for all numbers. Also be aware that the inequality may be true, it just is not a triangle inequality.

Which of these is a triangle inequality?

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Which of these is the definition of convergence of a sequence $(a_n)_{n \\in {\\mathbb N}}$ to $a$.

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$$
|a_n-a|<\\epsilon \\quad \\forall \\quad n \\ge N.
$$", "Given $N \\in {\\mathbb N}$ there exists $\\epsilon>0$ such that
$$
|a_n-a|<\\epsilon \\quad \\forall \\quad n \\ge N.
$$", "Given $N \\in {\\mathbb N}$ there exists $\\epsilon>0$ such that
$$
|a_n-a|<N \\quad \\forall \\quad n \\ge \\epsilon.
$$", "Given $N > 0$ there exists $\\epsilon \\in {\\mathbb N}$ such that
$$
|a_n-a|<N \\quad \\forall \\quad n \\ge \\epsilon.
$$", "Given $\\epsilon > 0$ for all $N \\in {\\mathbb N}$ such that
$$
|a_n-a|<\\epsilon \\quad \\exists \\quad n \\ge N.
$$", "Given $\\epsilon > 0$ there exists $N \\in {\\mathbb N}$ such that whenever $n \\ge {\\mathbb N}$
$$
|a_n-a|<\\epsilon.
$$"], "matrix": ["1", "-1", "-1", "1", "-1", "1"], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "The $\\epsilon$ $N$ game", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Jeremy Levesley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4981/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

In this question we will be testing your understanding of the following definition of convergence: Given $\\epsilon>0$ there exists $N \\in {\\mathbb N}$ such that
$$
|a_n-a|<\\epsilon \\quad \\forall \\quad n \\ge N.
$$

For each of the sequences below I will give you a value of $\\epsilon$ and you should give as small an appropriate $N$ as possible.

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Let $a_n = {\\var{c1} \\over n}$. Then $a_n \\rightarrow a=0$ as $n \\rightarrow \\infty$. If the value of $\\epsilon$ in the definition is $\\var{epsilon1}$ then the value of $N$ required so that $|a_n-a| < \\epsilon$ when $n \\ge N$ is [[0]].

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Let $a_n = {\\var{c2}n^{\\var{p}} +\\var{c4} \\over \\var{c3} n^{\\var{p}}}$. Then $a_n \\rightarrow a=\\var{c2}/\\var{c3}$ as $n \\rightarrow \\infty$. If the value of $\\epsilon$ in the definition is $\\var{epsilon2}$ then the value of $N$ required so that $|a_n-a| < \\epsilon$ when $n \\ge N$ is [[0]].

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This question will test your understanding of different properties of sequences, including monotonic, and bounded.

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The sequence $\\left ( {n-5 \\over 2n^2} \\right )_{n \\ge 1}$

is bounded below by 0 and above by 5.

", "

is monotone increasing.

", "

is monotone decreasing.

", "

converges to -5.

", "

converges to 0.

", "

is bounded below by -5 and above by 2.

", "

has an minimum.

", "

has a maximum.

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The sequence $\\left ( {n+5 \\over 2n} \\right )_{n \\ge 1}$

is bounded below by 1/2 and above by 3.

", "

is monotone increasing.

", "

is monotone decreasing.

", "

converges to 3.

", "

converges to 1/2.

", "

is bounded below by 0 and above by 2.

", "

has an infimum but no minimum.

", "

has a supremum but no maximum.

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This question will test your understanding of different properties of sequences and subsequences, including Cauchy sequences and the Bolzano Weierstrass theorem.

Which of the following statements are true about the sequence $a_n={(-1)^n \\over n}$, $n \\ge 1$.

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Which of the following statements are true about the sequence $a_n={3n-(-1)^n \\over n}$, $n \\ge 1$.

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This question tests your understanding of Cauchy sequences.

Which of these are Cauchy sequences?

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Which of these are Cauchy sequences?

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In this question we will test your ability to recognise if certain series converge.

Which of the following series converge?

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Which of the following series converge?

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In this question you will compute the radius of convergence of power series.

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What is the radius of convergence of $\\sum_{n=1}^\\infty n^{\\var{p1}} \\var{q1}^n x^n$?

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What is the radius of convergence of $\\sum_{n=1}^\\infty {1 \\over n^{\\var{p2}} } \\var{q2}^{-n} x^n$?

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What is the radius of convergence of $\\sum_{n=1}^\\infty \\exp(\\var{p3}n) x^{\\var{q3} n}$? Give your answer as a decimal to 2 significant figures.

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Let the functions $f$ and $g$ have limits $\\var{a}$ and $\\var{b}$ as $x \\rightarrow \\var{c}$ respectively.

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$\\lim_{x \\rightarrow \\var{c}} (f(x)+ g(x)) =$ [[0]].

$\\lim_{x \\rightarrow \\var{c}}{ f(x) \\over g(x)} =$ [[0]].

$\\lim_{x \\rightarrow \\var{c}} \\var{d} f(x)-g(x)=$ [[0]].

$\\lim_{n \\rightarrow \\infty} f(x)^\\var{p1}(g(x))^\\var{p2}=$ [[0]].

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This question tests your understanding of continuous functions.

Which of the following statements is true about a function $f$ which is continuous at $c \\in (a,b)$?

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Which of the following statements is true about functions $f$ and $g$ which are continuous at $c \\in (a,b)$?

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In this question you will decide which of the functions is continuous at the given point.

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Which of the functions $f$ is continous at the given point?

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f(x) = \\left \\{ \\begin{array}{cc}
\\var{p2}, & x < \\var{c2}, \\\\
-\\var{p2} & x \\ge \\var{c2},
\\end{array} \\right .
$$
at $x=\\var{c2}$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
\\var{p2}, & x < \\var{c2}, \\\\
-\\var{p2} & x \\ge \\var{c2},
\\end{array} \\right .
$$
at $x=\\var{c3}$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
x^\\var{p1}, & x < 0, \\\\
1-\\var{c1}x^\\var{p2} & x \\ge 0,
\\end{array} \\right .
$$
at $x=0$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
x^\\var{p1}, & x < 0, \\\\
1-\\var{c1}x^\\var{p2} & x \\ge 0,
\\end{array} \\right .
$$
at $x=\\var{c2}$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
\\var{c4}-\\sin(x), & x < 0, \\\\
1-\\var{c1}\\cos(x) & x \\ge 0,
\\end{array} \\right .
$$
at $x=0$."], "matrix": ["1", "-1", "1", "-1", "1", "1"], "distractors": ["", "", "", "", "", ""]}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Which of the functions $f$ is continous at the given point?

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f(x) = \\left \\{ \\begin{array}{cc}
\\var{p2}, & x < \\var{c1}, \\\\
-\\var{p2} & x \\ge \\var{c1},
\\end{array} \\right .
$$
at $x=\\var{c1}$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
\\var{p2}, & x < \\var{c1}, \\\\
-\\var{p2} & x \\ge \\var{c1},
\\end{array} \\right .
$$
at $x=\\var{c5}$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
2+x^\\var{p1}, & x < 0, \\\\
2-\\var{c1}x^\\var{p2} & x \\ge 0,
\\end{array} \\right .
$$
at $x=0$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
(x-\\var{c2})^\\var{p1}, & x < \\var{c2}, \\\\
1-\\var{c1}(x-\\var{c2})^\\var{p2} & x \\ge \\var{c2},
\\end{array} \\right .
$$
at $x=\\var{c2}$.", "$$
f(x) = \\left \\{ \\begin{array}{cc}
\\var{c2p}-\\sin(x), & x < 0, \\\\
1+\\var{c2pm}\\cos(x) & x > 0, \\\\
\\var{c2p}, & x=0.
\\end{array} \\right .
$$
at $x=0$."], "matrix": ["1", "-1", "1", "1", "-1", "1"], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "warnifunattempted", "message": "

Remember to come back and answer this question.

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You have 5 minutes left.

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