MCV4U V.1
UNIT 1 TEST: INTRODUCTION TO CALCULUS
Name:
COMMUNICATION
/6 + /4
Date:
COMMUNCATION - TEST
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/12
APPLICATION
THINKING
/12
/11
(2 MARKS)
Enough steps shown to clearly demonstrate thinking
Solutions that are neat and easy to follow
Proper use of mathematical symbols and notation
Equal signs (aligned, one per line)
Side calculations clearly labeled if used
Units used as required
Variables and functions defined as required
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COMMUNCATION - UPLOAD
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KNOWLEDGE
Concluding statements used as required Fractions, not
decimals, unless otherwise stated
Radicals, not decimals, unless otherwise stated
Fractions reduced to lowest terms
Denominators rationalized
Labels, arrows, smooth lines, and ruler used as required
Labelled axis and titled graph
(2 MARKS)
Units used as required
All work submitted as one PDF file (-1 mark)
PDF file follows proper naming convention
LastName_FirstName_Chapter#_Test/Quiz
PDF files must be in portrait mode
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PDF file must be submitted in the appropriate assignment
/ quiz location. (-2 marks)
PDF must contain the full page
PDF is in order (-1 mark)
PDF must be clear and readable (-2 marks)
UNSUPPORTED WORK WILL RECEIVE A MARK OF ZERO FOR THAT QUESTION.
COMMUNICATION:
1. Explain the following statement: A function does not need to be continuous at the point x=a in order
for the limit to exist at the point x=a. Provide an equation and a sketch of such a function.
(4)
2. The limit below represents the derivative of some function f(x) evaluated at some point x  a
Determine the function f(x) and the value of a.
2(6 + ℎ)2 − 𝑠(6)2
lim
ℎ→0
ℎ
(2)
KNOWLEDGE:
3. Use the graph to estimate the limits and value of the function, or explain why the limits do not exist.
lim 𝑓(𝑥)
𝑥→2+
lim 𝑓(𝑥)
lim 𝑓(𝑥)
𝑥→2−
lim 𝑓(𝑥)
𝑥→1−
𝑥→3
lim 𝑓(𝑥)
𝑓(2)
𝑥→2
(4)
State any points of discontinuity.
4. Evaluate each of the following limits.
a)
𝑥 2 −𝑎2
,
𝑥→𝑎 √𝑥−√𝑎
lim
𝑎>0
b)
lim ((4𝑥 −
𝑥→3
2
) (6 +
𝑥−3
𝑥 − 𝑥 2 ))
(4)
5. Find the equation of the tangent line to the curve 𝑦 = √𝑥 + 1 at 𝑥 = 3. Graph the function and the
tangent line. Label the function and tangent line on the graph. No decimals.
(4)
APPLICATION:
6. Let
𝑥 , 𝑖𝑓 𝑥 < 1
3 , 𝑖𝑓 𝑥 = 1
𝑔(𝑥) =
2 − 𝑥 2 , 𝑖𝑓 1 < 𝑥 ≤ 2
3, , 𝑖𝑓 𝑥 > 2
{
Evaluate each limit, if it exits.
a) lim− 𝑔(𝑥)
𝑥→1
(3)
b) lim 𝑔(𝑥)
𝑥→1
c)
lim 𝑔(𝑥)
𝑥→1+
7. For what values of A and B is this statement correct? Justify your response.
√𝐴𝑥 + 𝐵 − 3
lim
=1
𝑥→0
𝑥
(2)
8. A carpenter is constructing a large cubical storage shed. The volume of the shed is given by
16−𝑥 2
𝑉(𝑥) = 𝑥 2 (
), where x is the side length in metres. Determine the instantaneous rate of
4𝑥
change of volume when the side length is 2 metres.
(3)
9. Sketch the graph of a function g x  , such that the following characteristics are satisfied:
lim 𝑔(𝑥) = −2
𝑥→2−
lim 𝑔(𝑥) = 0
𝑥→−2+
lim 𝑔(𝑥) = 3
𝑥→0+
lim 𝑔(𝑥) = +∞
𝑥→3+
lim 𝑔(𝑥) = 0
𝑥→−∞
(4)
𝑔(−2) = 0
lim 𝑔(𝑥) = 3
𝑥→0−
lim 𝑔(𝑥) = +∞
𝑥→3−
lim 𝑔(𝑥) = +∞
𝑥→+∞
THINKING:
10. For what values of c is the function 𝑔(𝑥) continuous at every number?
(𝑐𝑥 − 1)3 ,
𝑔(𝑥) = { 2 2
𝑐 𝑥 − 1,
−∞ < 𝑥 < 2
2≤𝑥<∞
(3)
11. Sketch the function of 𝑓(𝑥), and determine lim 𝑓(𝑥).
𝑥→1
Show all you work.
2 + 𝑥,
𝑓(𝑥) = {1,
𝑥 − 1,
𝑥<1
𝑥=1
𝑥>1
(4)
12. Determine the 4 tiered piecewise equation of the
following function. (Question 11 is a 3 tiered piecewise
function).
(4)