MCV4U · Unit 3 test review

Curve Sketching: Test Review

Everything for the Unit 3 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Increasing and Decreasing Functions and the First Derivative Test

If $f^{'} (x) > 0$ on an interval, $f$ is increasing there; if $f^{'} (x) < 0$ , $f$ is decreasing there.
A critical number is a $c$ in the domain of $f$ with $f^{'} (c) = 0$ or $f^{'} (c)$ undefined.
First derivative test: $f^{'}$ changes $+$ to $-$ at $c$ gives a local maximum; $-$ to $+$ gives a local minimum; no sign change gives neither.
$f^{'} (c) = 0$ alone does not guarantee an extremum: $f (x) = x^{3}$ has $f^{'} (0) = 0$ but no maximum or minimum at $0$ .
Find the height of an extremum from $f (c)$ , not $f^{'} (c)$ .

Concavity, Points of Inflection and the Second Derivative Test

$f^{''} (x)$ is the derivative of $f^{'} (x)$ : the rate of change of the slope of the tangent.
$f^{''} (x) > 0$ means concave up (slopes increasing); $f^{''} (x) < 0$ means concave down (slopes decreasing).
A point of inflection is a point on the graph where the concavity changes; check that $f^{''}$ changes sign, since $f^{''} (c) = 0$ alone is not enough (e.g. $x^{4}$ at $0$ ).
At an inflection point where $f^{'}$ exists, $f^{'}$ has a local maximum or minimum.
Second derivative test: $f^{'} (c) = 0$ and $f^{''} (c) > 0$ gives a local minimum; $f^{''} (c) < 0$ gives a local maximum; $f^{''} (c) = 0$ is inconclusive.

Sketching Polynomial and Simple Rational Functions

Sketching order: domain, intercepts, symmetry, $f^{'}$ (critical numbers, increase and decrease, extrema), $f^{''}$ (concavity, inflection points), asymptotes and end behaviour, then sketch and verify.
Even: $f (- x) = f (x)$ , symmetric about the $y$ -axis. Odd: $f (- x) = - f (x)$ , symmetric about the origin.
A degree- $n$ polynomial has at most $n - 1$ turning points and at most $n - 2$ points of inflection.
For a rational function, vertical asymptotes go on the sign charts as boundaries, but they are never critical numbers or inflection points.
Horizontal asymptote: $y = 0$ if the degree of the numerator is smaller; the ratio of leading coefficients if the degrees are equal.

Graphs of f, f′ and f″

$f^{'} > 0$ on an interval $\to$ $f$ increasing (a rising $f$ can still have $f^{'} = 0$ at a single flat point); local max or min of $f$ $\leftrightarrow$ $f^{'}$ crosses zero ( $+$ to $-$ or $-$ to $+$ ).
$f^{''} > 0$ on an interval $\to$ $f^{'}$ increasing $\leftrightarrow$ $f$ concave up (a concave-up $f$ can still have $f^{''} = 0$ at a single point, like $x^{4}$ at $0$ ).
Inflection point of $f$ $\leftrightarrow$ local max or min of $f^{'}$ $\leftrightarrow$ $f^{''}$ changes sign.
$f (x) + k$ has the same derivative as $f (x)$ ; the derivative of $f (x - h)$ is $f^{'} (x - h)$ .
Knowing $f^{'}$ determines $f$ only up to a vertical shift: $f (x) + C$ for any constant $C$ . One known point fixes $C$ .

Where marks get lost

Classifying every solution of $f^{'} (x) = 0$ as a maximum or minimum without checking the sign on both sides. In $x^{4} - 4 x^{3}$ , $x = 0$ is neither. (Increasing and Decreasing Functions and the First Derivative Test)
Substituting critical numbers into $f^{'} (x)$ to find the height of a turning point. $f^{'} (c) = 0$ by definition; the height is $f (c)$ . (Increasing and Decreasing Functions and the First Derivative Test)
Solving $f (x) = 0$ (the $x$ -intercepts) instead of $f^{'} (x) = 0$ (the horizontal tangents) when looking for turning points. (Increasing and Decreasing Functions and the First Derivative Test)
Forgetting the points where $f^{'} (x)$ does not exist, such as the cusp of $x^{\frac{2}{3}}$ at $x = 0$ . (Increasing and Decreasing Functions and the First Derivative Test)
Writing the intervals of increase as $y$ -values, or including a critical number in two intervals as if it were a whole interval. State intervals of $x$ , such as $(- \infty, - 1)$ and $(2, \infty)$ . (Increasing and Decreasing Functions and the First Derivative Test)
Calling every solution of $f^{''} (x) = 0$ an inflection point. For $f (x) = x^{4}$ , $f^{''} (0) = 0$ but the concavity does not change, so there is no inflection point. (Concavity, Points of Inflection and the Second Derivative Test)
Mixing up the second derivative test: a POSITIVE $f^{''} (c)$ means concave up, a cup shape, so a MINIMUM (not a maximum). (Concavity, Points of Inflection and the Second Derivative Test)
Concluding "no extremum" when $f^{''} (c) = 0$ . The test is only inconclusive; $x^{4}$ has a minimum at $0$ even though $f^{''} (0) = 0$ . (Concavity, Points of Inflection and the Second Derivative Test)
Thinking concave up means increasing. $y = x^{2}$ is concave up everywhere but decreasing for $x < 0$ . (Concavity, Points of Inflection and the Second Derivative Test)
Giving an inflection point as an $x$ -value only, or calling a concavity change at a vertical asymptote an inflection point. A point of inflection is a point $(x, f (x))$ on the graph. (Concavity, Points of Inflection and the Second Derivative Test)
Trusting a single calculator window. A turning point outside the window, or two close together, is easy to miss; solve $f^{'} (x) = 0$ to be sure you have them all. (Sketching Polynomial and Simple Rational Functions)
Calling a function odd because its leading term has an odd degree. $x^{3} + 1$ is neither even nor odd: the constant term breaks the symmetry. (Sketching Polynomial and Simple Rational Functions)
Listing a vertical asymptote as a critical number or an inflection point. The function is not defined there, so it cannot have a point of the graph there. (Sketching Polynomial and Simple Rational Functions)
Leaving vertical asymptotes off a sign chart, then reading a sign that is wrong on one side of the asymptote. (Sketching Polynomial and Simple Rational Functions)
Drawing a cubic with no inflection point or a quartic with four turning points. Count: degree $n$ allows at most $n - 1$ turning points. (Sketching Polynomial and Simple Rational Functions)
Reading the HEIGHT of the graph of $f^{'}$ as the height of $f$ . The highest point of the graph of $f^{'}$ marks the steepest uphill point of $f$ , not its maximum. (Graphs of f, f′ and f″)
Putting the maximum of $f$ where $f^{'}$ has its maximum. The maximum of $f$ is where $f^{'}$ crosses zero going from positive to negative. (Graphs of f, f′ and f″)
Claiming there is only one function with a given derivative. Any vertical shift $f (x) + C$ has the same derivative, so you need a point such as $f (0)$ to pin it down. (Graphs of f, f′ and f″)
Thinking a vertical shift of $f$ shifts the graph of $f^{'}$ . Shifting up or down changes no slopes; of the two kinds of translation, only a horizontal shift moves the graph of $f^{'}$ . (Graphs of f, f′ and f″)
Sketching $f^{'}$ with the same degree as $f$ . Differentiating a polynomial lowers the degree by one, so a cubic's derivative is a parabola. (Graphs of f, f′ and f″)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Increasing and Decreasing Functions and the First Derivative Test
1
On the interval $(2, 5)$ , $f^{'} (x) > 0$ . What does this tell you about $f$ ?
Increasing and Decreasing Functions and the First Derivative Test
2
$f (x) = x^{2} + 3 x - 10$ has one critical number. What is it? Give a decimal.
Increasing and Decreasing Functions and the First Derivative Test
3
How many critical numbers does $f (x) = x^{4} - 8 x^{2} + 7$ have?
Increasing and Decreasing Functions and the First Derivative Test
4
For $f (x) = x^{3}$ , $f^{'} (0) = 0$ . What happens at $x = 0$ ?
Increasing and Decreasing Functions and the First Derivative Test
5
Match each piece of information about $f^{'}$ with what it tells you about $f$ .
1. A. $f$ has a local maximum at $c$
2. B. $f$ has a local minimum at $c$
3. C. $f$ is decreasing on that interval
4. D. $f$ is increasing on that interval
$f^{'} (x) > 0$ on an interval
$f^{'} (x) < 0$ on an interval
$f^{'}$ changes from positive to negative at $c$
$f^{'}$ changes from negative to positive at $c$
Increasing and Decreasing Functions and the First Derivative Test
6
What is the name for a number $c$ in the domain of $f$ where $f^{'} (c) = 0$ or $f^{'} (c)$ does not exist? (Two words.)
Concavity, Points of Inflection and the Second Derivative Test
7
On an interval, $f^{''} (x) > 0$ . Which statement must be true there?
Concavity, Points of Inflection and the Second Derivative Test
8
Find the $x$ -coordinate of the point of inflection of $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ .
Concavity, Points of Inflection and the Second Derivative Test
9
For $f (x) = x^{4}$ , $f^{''} (0) = 0$ . Is $(0, 0)$ a point of inflection?
Concavity, Points of Inflection and the Second Derivative Test
10
A function has $f^{'} (4) = 0$ and $f^{''} (4) = - 5$ . What can you conclude?
Concavity, Points of Inflection and the Second Derivative Test
11
Match each condition with its conclusion.
1. A. $(c, f (c))$ is a point of inflection
2. B. $f$ has a local minimum at $c$
3. C. $f$ is concave down there
4. D. $f$ is concave up there
$f^{''} (x) > 0$ on an interval
$f^{''} (x) < 0$ on an interval
$f^{''}$ changes sign at $c$ , which is in the domain of $f$
$f^{'} (c) = 0$ and $f^{''} (c) > 0$
Concavity, Points of Inflection and the Second Derivative Test
12
What is a point on a graph called where the concavity changes? (Accepts the usual names.)
Sketching Polynomial and Simple Rational Functions
13
Which function is even?
Sketching Polynomial and Simple Rational Functions
14
An odd function has a local maximum at $(- 2, 16)$ . What is the $y$ -coordinate of the matching local minimum at $x = 2$ ?
Sketching Polynomial and Simple Rational Functions
15
Find the maximum value of $f (x) = \frac{4 x}{x ^{2} + 1}$ .
Sketching Polynomial and Simple Rational Functions
16
What is the greatest number of turning points the graph of a quartic (degree 4) polynomial can have?
Sketching Polynomial and Simple Rational Functions
17
Match each feature with how you find it.
1. A.Evaluate $f (0)$
2. B.Sign chart of $f^{''} (x)$
3. C.Sign chart of $f^{'} (x)$
4. D.Solve $f (x) = 0$
5. E.Zeros of the denominator that do not cancel
$y$ -intercept
$x$ -intercepts
Intervals of increase and decrease
Intervals of concavity
Vertical asymptotes of a rational function
Sketching Polynomial and Simple Rational Functions
18
What is the horizontal asymptote of $f (x) = \frac{4 x}{x ^{2} + 1}$ ?
Graphs of f, f′ and f″
19
At a local maximum of a smooth function $f$ , what does the graph of $f^{'}$ do?
Graphs of f, f′ and f″
20
If $g (x) = f (x) + 5$ , how does the graph of $g^{'}$ compare with the graph of $f^{'}$ ?
Graphs of f, f′ and f″
21
How many different functions have derivative $f^{'} (x) = 2 x - 4$ ?
Graphs of f, f′ and f″
22
$f^{'} (x) = 2 x - 4$ and the graph of $f$ passes through $(0, 1)$ . What is the minimum value of $f$ ?
Graphs of f, f′ and f″
23
$f^{'} (x) = 3 (x + 2) (x - 4)$ . At what $x$ -value does $f$ have a local maximum?
Graphs of f, f′ and f″
24
Match each feature of the graph of $f^{'}$ with the matching feature of $f$ .
1. A. $f$ is concave up
2. B. $f$ is decreasing
3. C.Local minimum of $f$
4. D.Point of inflection of $f$ (concave up to concave down)
$f^{'}$ crosses the axis from $-$ to $+$
$f^{'}$ has a local maximum
$f^{'}$ is below the axis
$f^{'}$ is increasing

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…