MCV4U · Unit 1 test review

Rates of Change & Limits: Test Review

Everything for the Unit 1 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

Average and Instantaneous Rates of Change

Rate of change $= \frac{Δ y}{Δ x}$ , with units of output units per input unit (m/s, L/min, °C/km).
Average rate of change from $a$ to $b$ : $\frac{f ( b ) - f ( a )}{b - a}$ , the slope of the secant through $(a, f (a))$ and $(b, f (b))$ .
Instantaneous rate of change at $x = a$ : the slope of the tangent at $(a, f (a))$ .
Estimate a tangent slope with secants through the point: shrink the interval from both sides and see what value the slopes approach.
A graph with a corner at a point has no tangent there, so its instantaneous rate of change at that point does not exist.

Limits: Graphs, Tables, Sequences and Continuity

$lim_{x \to a} f (x) = L$ means $f (x)$ gets as close to $L$ as you like when $x$ is close enough to $a$ (but not equal to it).
The two-sided limit exists only if $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) .$
If substitution gives $\frac{0}{0}$ , simplify (factor and cancel) and then substitute.
Famous limits: $(1 + \frac{1}{n})^{n} \to e \approx 2.718$ , Fibonacci ratios $\to φ \approx 1.618$ , $4 (1 - \frac{1}{3} + \frac{1}{5} - \dots) \to π$ .
$f$ is continuous at $a$ if $f (a)$ is defined, the limit exists, and $lim_{x \to a} f (x) = f (a)$ .

The Derivative from First Principles

Difference quotient: $\frac{f ( a + h ) - f ( a )}{h}$ , the slope of the secant from $x = a$ to $x = a + h$ (an average rate of change).
Derivative at a point: $f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h},$ the slope of the tangent (an instantaneous rate of change).
You cannot substitute $h = 0$ directly ( $\frac{0}{0}$ ); expand, simplify and cancel the factor of $h$ first.
For $f (x) = x^{2}$ at $x = 3$ : the quotient simplifies to $6 + h$ , so $f^{'} (3) = 6$ .
$f^{'} (a)$ has units of (output units) per (input unit), such as m/s.

The Derivative Function

Derivative function: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h};$ it gives the tangent slope at every $x$ .
Notation: $f^{'} (x)$ , $y^{'}$ , $\frac{d y}{d x}$ and $\frac{d}{d x} f (x)$ all mean the derivative.
For polynomials the degree drops by one: linear gives constant, quadratic gives linear, cubic gives quadratic.
Where $f$ rises, $f^{'} (x) > 0$ ; where $f$ falls, $f^{'} (x) < 0$ (apart from single flat points, like $x^{3}$ at $0$ ); a smooth local maximum or minimum has $f^{'} (x) = 0$ .
No derivative at a corner, cusp, discontinuity or vertical tangent. Differentiable implies continuous, but not the reverse.

Where marks get lost

Mixing the order in the slope formula, such as $\frac{f ( b ) - f ( a )}{a - b}$ . Subtract in the same order on top and bottom, or the sign of the rate flips. (Average and Instantaneous Rates of Change)
Finding the average of the function's VALUES instead of the average RATE. The average rate over an interval only uses the two endpoints: $\frac{f ( b ) - f ( a )}{b - a}$ . (Average and Instantaneous Rates of Change)
Leaving off units or using the wrong ones. A volume in litres changing over minutes has a rate in L/min, not L or min. (Average and Instantaneous Rates of Change)
Estimating a tangent slope from one side only with a large interval. Use small intervals, check both sides, and look for the value the slopes approach. (Average and Instantaneous Rates of Change)
Assuming every graph has a tangent everywhere. At a corner the left and right secants approach different slopes, so the instantaneous rate there does not exist. (Average and Instantaneous Rates of Change)
Concluding that a limit does not exist just because $f (a)$ is undefined. The hole in $\frac{x ^{2} - 1}{x - 1}$ at $x = 1$ still has a limit of $2$ . (Limits: Graphs, Tables, Sequences and Continuity)
Stopping at $\frac{0}{0}$ and calling the answer $0$ or undefined. $\frac{0}{0}$ is a signal to simplify, not an answer. (Limits: Graphs, Tables, Sequences and Continuity)
Checking only one side. If the left-hand and right-hand limits are different, the two-sided limit does not exist. (Limits: Graphs, Tables, Sequences and Continuity)
Assuming a limit must equal the function's value. They agree only where the function is continuous. (Limits: Graphs, Tables, Sequences and Continuity)
Deciding a sequence's limit from the first few terms. The partial sums $4 (1 - \frac{1}{3} + \dots)$ start at $4$ and $2.67$ , nowhere near the limit $π$ ; look at the long-run trend. (Limits: Graphs, Tables, Sequences and Continuity)
Writing $f (a + h)$ as $f (a) + h$ . For $f (x) = x^{2}$ , $f (3 + h) = (3 + h)^{2} = 9 + 6 h + h^{2}$ , not $9 + h$ . (The Derivative from First Principles)
Expanding $(3 + h)^{2}$ as $9 + h^{2}$ . The middle term $6 h$ is the one that survives the limit, so dropping it loses the answer. (The Derivative from First Principles)
Substituting $h = 0$ before simplifying, getting $\frac{0}{0}$ , and concluding the derivative is $0$ or undefined. (The Derivative from First Principles)
Forgetting to distribute a negative or a coefficient: $- 5 (3 + h) = - 15 - 5 h$ , not $- 15 + h$ or $- 15 - h$ . (The Derivative from First Principles)
Stopping at the simplified quotient and giving $7 + 2 h$ as the answer. The derivative at $x = a$ is a number: the final step must actually let $h \to 0$ . (The Derivative from First Principles)
Substituting a number for $x$ before taking the limit when the question asks for $f^{'} (x)$ . Keep $x$ as a variable; substitute only at the end, if a slope at a point is wanted. (The Derivative Function)
Reading the HEIGHT of $f$ as the height of $f^{'}$ . Where $f$ is high but flat, $f^{'}$ is near $0$ ; where $f$ is below the axis but rising, $f^{'}$ is positive. (The Derivative Function)
Thinking a zero of $f$ is a zero of $f^{'}$ . The zeros of $f^{'}$ are where the tangent is horizontal (peaks, valleys and flat spots), not where $f$ crosses the $x$ -axis. (The Derivative Function)
Assuming every continuous function is differentiable. The graph of $y = ∣ x ∣$ is unbroken, but it has no tangent slope at the corner $x = 0$ . (The Derivative Function)

Practice test

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

Average and Instantaneous Rates of Change
1
Find the average rate of change of $f (x) = x^{3} - 2 x$ from $x = 1$ to $x = 3$ .
Average and Instantaneous Rates of Change
2
Using the tank table ( $V = 420$ L at $t = 2$ min and $V = 296$ L at $t = 6$ min), find the average rate of change of the volume from $t = 2$ to $t = 6$ , in L/min.
L/min
Average and Instantaneous Rates of Change
3
Secants of a smooth curve through the point $x = 5$ have slopes $3.2$ (for $h = 0.1$ ), $3.02$ ( $h = 0.01$ ), $2.98$ ( $h = - 0.01$ ) and $2.8$ ( $h = - 0.1$ ). What is the best estimate of the instantaneous rate of change at $x = 5$ ?
Average and Instantaneous Rates of Change
4
What is the name of the line whose slope gives the instantaneous rate of change at a point on a smooth curve?
Average and Instantaneous Rates of Change
5
Why can the instantaneous rate of change of $f (x) = ∣ x - 2∣ + 1$ not be determined at $x = 2$ ?
Average and Instantaneous Rates of Change
6
Find the average rate of change of $g (x) = 2^{x}$ from $x = 0$ to $x = 3$ . Give your answer to 2 decimal places.
Limits: Graphs, Tables, Sequences and Continuity
7
Evaluate $lim_{x \to 4} \frac{x ^{2} - 16}{x - 4}$ .
Limits: Graphs, Tables, Sequences and Continuity
8
The graph of $f$ is the line $y = x + 1$ with an open circle at $(1, 2)$ and a separate filled dot at $(1, 4)$ . Which statement is true?
Limits: Graphs, Tables, Sequences and Continuity
9
The sequence $(1 + \frac{1}{1})^{1}, (1 + \frac{1}{2})^{2}, (1 + \frac{1}{3})^{3}, \dots$ approaches which famous number? (Give its symbol or name.)
Limits: Graphs, Tables, Sequences and Continuity
10
Match each sequence to its limit.
1. A. $π \approx 3.142$
2. B. $φ \approx 1.618$
3. C. $0$
4. D. $e \approx 2.718$
$(1 + \frac{1}{n})^{n}$
Ratio of consecutive Fibonacci numbers
Partial sums of $4 (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots)$
$\frac{1}{n}$
Limits: Graphs, Tables, Sequences and Continuity
11
What kind of discontinuity does $f (x) = \frac{x ^{2} - 1}{x - 1}$ have at $x = 1$ ?
Limits: Graphs, Tables, Sequences and Continuity
12
Evaluate $lim_{x \to - 2} (x^{2} + 3 x)$ .
The Derivative from First Principles
13
Use first principles to find the slope of the tangent to $f (x) = x^{2} - 3 x$ at $x = 5$ .
The Derivative from First Principles
14
For $f (x) = x^{2}$ at $x = 3$ , the difference quotient simplifies to $6 + h$ . What is the slope of the secant from $x = 3$ to $x = 3.05$ ?
The Derivative from First Principles
15
Why can you not find $lim_{h \to 0} \frac{( 3 + h ) ^{2} - 9}{h}$ by substituting $h = 0$ right away?
The Derivative from First Principles
16
Use first principles to find $f^{'} (- 1)$ for $f (x) = 4 - x^{2}$ .
The Derivative from First Principles
17
Match each expression to what it represents for the graph of $y = f (x)$ .
$h$
$f (a + h) - f (a)$
$\frac{f ( a + h ) - f ( a )}{h}$
$lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$
The Derivative from First Principles
18
A ball's height is $s (t) = 30 t - 5 t^{2}$ metres after $t$ seconds. Use first principles to find its velocity at $t = 2$ , in m/s.
m/s
The Derivative Function
19
If $f (x)$ is a cubic polynomial, what kind of function is $f^{'} (x)$ ?
The Derivative Function
20
Match each function to its derivative function.
1. A. $f^{'} (x) = 0$
2. B. $f^{'} (x) = 2 x$
3. C. $f^{'} (x) = 3 x^{2}$
4. D. $f^{'} (x) = 4$
$f (x) = 7$
$f (x) = 4 x - 1$
$f (x) = x^{2}$
$f (x) = x^{3}$
The Derivative Function
21
Which function is NOT differentiable at $x = 0$ ?
The Derivative Function
22
The graph of $y = ∣ x - 1∣$ has a sharp point at $x = 1$ where the left and right slopes are different. What is this kind of point called?
The Derivative Function
23
For $f (x) = 4 - x^{2}$ , put the tangent slopes at these points in order from smallest to largest.
1. 1At $x = 1$
2. 2At $x = - 2$
3. 3At $x = 0$
4. 4At $x = 3$
The Derivative Function
24
For $f (x) = - 3 x + 7$ , what is $f^{'} (100)$ ?

Endless drill

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

Endless drillnew numbers every time

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