CEGEP Calculus I — Analysis of Functions and Curve Sketching

All 8 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Domain, Zeros and Intercepts

   Determine the domain, the zeros and the $y$-intercept of each function. Where a feature does not exist, say so and explain why.

   1. ${\displaystyle f(x)=\frac{x^3-4x}{x^2-2x-3}}$
   2. $h(x)=\sqrt{x+1}\ln x$
2. Q2Locating Vertical and Horizontal Asymptotes

   Let ${\displaystyle f(x)=\frac{3x^2-12}{x^2-x-6}}$. Find every vertical and every horizontal asymptote of the graph of $f$, and justify each one with a limit. Identify the point where the denominator is zero but no vertical asymptote occurs, and give its coordinates.

3. Q3Increasing and Decreasing Intervals

   Let $f(x)=x^2{e}^{-x}$. Find $f^{\prime }$, then determine every interval on which $f$ is increasing and every interval on which it is decreasing, supporting your answer with a sign analysis of $f^{\prime }$.

4. Q4Relative and Absolute Extrema

   Let $f(x)=2x^3-9x^2+12x$ on the closed interval $[0,3]$.

   1. Find the critical numbers of $f$ and classify each one with the first derivative test.
   2. Find the absolute maximum and the absolute minimum of $f$ on $[0,3]$, stating where each occurs.
5. Q5Concavity and Points of Inflection

   Let $f(x)=x^4-4x^3+5$. Determine the intervals on which the graph of $f$ is concave up and those on which it is concave down, and give the coordinates of every point of inflection.

6. Q6Sketching a Complete Curve

   Carry out a complete study of ${\displaystyle f(x)=\frac{x^2}{x^2-4}}$ — domain, intercepts, symmetry, vertical and horizontal asymptotes with the relevant limits, intervals of increase and decrease with any relative extrema, and intervals of concavity with any points of inflection — then sketch the curve on the grid, labelling each asymptote.

   A blank Cartesian grid for this question is on the printable PDF.

7. Q7Rolle's Theorem and the Mean Value Theorem

   Name each hypothesis as you check it.

   1. Verify that $f(x)=x^3-4x$ satisfies the hypotheses of Rolle's Theorem on $[-2,2]$, then find every value of $c$ the theorem guarantees.
   2. Verify that $g(x)=\sqrt{x}$ satisfies the hypotheses of the Mean Value Theorem on $[1,9]$, then find every value of $c$ the theorem guarantees.
8. Q8Synthesis — drawing on several topics in this set

   Let ${\displaystyle f(x)=\frac{6x}{x^2+3}}$. Carry out a full analysis:

   1. domain, zeros, $y$-intercept and symmetry;
   2. every vertical and horizontal asymptote, justified by a limit;
   3. intervals of increase and decrease, and every relative extremum;
   4. the absolute maximum and absolute minimum values of $f$ on $ℝ$, with a reason — note that the Extreme Value Theorem does not apply here;
   5. intervals of concavity and every point of inflection;
   6. a short description of the resulting shape.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The ministerial list is the method

The devis does not ask for a drawing. It asks for an étude d'une fonction, and it names the parts in order. Q8 is that whole list carried out on a single function; Q1 to Q7 are its parts practised one at a time. Run the list in the same order on every function you are given and the sketch stops being an act of drawing — it becomes a summary of what you have already proved.

Domain first, because everything after it lives inside the domain

Q1 looks like the easy question on the sheet, and it is the one that quietly decides the rest. A rational rule excludes the zeros of its denominator. A square root needs its radicand to be at least zero. A logarithm needs its argument strictly positive. When a rule combines two of these — as the second function in Q1 does — the conditions have to hold at the same time, so you intersect them rather than list them.

A zero of the denominator is not automatically a vertical asymptote

This is the whole point of Q2, which asks you to find the value where the denominator vanishes and no asymptote appears. Factor the numerator and the denominator before you conclude anything, and cancel what cancels — while noting that the cancelled value stays out of the domain regardless.

Critical numbers: the half of the definition that gets dropped

A critical number of f is a value c in the domain of f at which either f’(c) = 0 or f’(c) does not exist. Setting the derivative equal to zero finds some critical numbers. It does not find all of them.

Relative is a local claim; absolute is a claim about everything

Q4 separates the two on purpose. Part (a) classifies the critical numbers with the first derivative test — f’ changing from positive to negative gives a relative maximum, negative to positive gives a relative minimum, and no sign change gives neither. Part (b) then asks a different question on a closed interval, and the answer to (a) does not settle it.

Q8 asks for the absolute extrema on all of the reals, where the Extreme Value Theorem has nothing to say — its hypothesis of a closed, bounded interval fails, and the question tells you so. The argument there comes from the monotonic behaviour instead: if f decreases on one side, rises to a value, and decreases after it, then no value of f can exceed that one. That is a complete proof, and it needs no theorem to license it.

Concavity, inflection, and the asymptote that fakes one

Q5 is the second sign chart, run on f’’. Concave up where f’’ > 0, concave down where f’’ < 0, and a point of inflection where the sign changes. A zero of f’’ is only a candidate: it marks where to look, and the sign change is the fact.

Rolle and the Mean Value Theorem: name each hypothesis as you check it

Q7 says so in its instruction, because at this level the check is the answer. A theorem applies when its hypotheses hold, and writing down which ones hold and why is the rigorous reasoning the competency asks for.

The sketch has to agree with the work above it

Q6 and Q8 both end in a curve, and the curve is judged against the analysis rather than against how smooth it looks. Draw every asymptote dashed first, then plot only the points you actually computed — intercepts, relative extrema, points of inflection — label them with their coordinates, and join them with arcs that bend the way your second sign chart says they do.

Getting the most out of it

Run the list, in the same order, every time

Domain, zeros and intercept, symmetry, asymptotes, f’, f’’, sketch. Write those headings down your page before you compute anything, and fill them in. The order is not a preference — it is the order in which each step supplies what the next one needs, and it is the order the devis lists.

Factor before you conclude anything

Almost every trap in this material is disarmed by factoring first: the difference between a hole and a vertical asymptote, the zeros that turn out not to be in the domain, the sign of a derivative you would otherwise have to multiply out. A factored expression hands you its zeros and the sign of each piece for nothing.

Justify from the derivative, not from the picture

“f is increasing on (0, 2) because f’(x) > 0 there” earns the mark; “the graph goes up” does not. When you use a named theorem, name it and check its hypotheses in the same breath. When a theorem does not apply, say which hypothesis fails and where.

Ask whether the interval is closed, every single time

Closed and bounded: the Extreme Value Theorem applies, the extrema are guaranteed to exist, and the endpoints join the list of candidates. Open, or all of the reals: no guarantee, no endpoints, and you argue from monotonic behaviour or from limits instead. A large share of the avoidable marks lost in this material is the wrong branch of that one question.

Finish by checking the sketch against your own sentences

Read your conclusions back one at a time and look at the curve while you do it. Decreasing here, concave up there, this asymptote approached from above. Anything the picture and the prose disagree about is an error you can still fix.

Common questions