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MCV4U Name:UNIT 1 TEST
THE DERIVATIVE
V;
TH/10
COMM
An overall comrptlhication mark will be assigned for proper use of mathematicalform, termipeiogy and conventions. DERIVATIVE RULES MAY BE USED
EXCEPT WHERE LIMIT THEORY IS SPECIFIED.
78A: KNOWLEDGE/UNDERSTANDING
1. Determine the value of each of the following limits if it exists.
, . 3-VX + 9 5 .i \JbJ~rib) ton .a) lim—=
' ~2.52JC
•= \ i tvvY-3!
c) lim —-* -. V „,-, I I*^JC
\n8
r-2
,
2. a) Determine the slope of the tangent to f(x) - -x3 at x = -1
b ) Determine the slope of the tangent to /(*) = — at x = 2
-Z
m -
3. Determine — for each of the following. Simplify.dx
a) y =
7"
4. A subway train travels from one station to the next. Its distance, inkilometres, from the first station after t mins is s(t] = t2 -\t3.
a) Find the average velocity of the train between t = 0 and t = 1.
= o-6- \ -\
b) Find the velocity of the train at 90 seconds. ^/
1V\<*N
c) Show that the next station the train stops at is 2-tern, away., •£.
B: APPLICATION5. a) Use the Product rule to determine the rate of change of f(x) = x2 (3x -1)
at x=3. ^> /3
b) Show that you get the same result using the limit definition of the derivative.
if 14
X"
=. \irrs
c) Show another method that could be used to verify the value of /'(3)
/3
6. Determine the points on the curvehorizontal.
- \-I
where the tangent line is
/5
m * o
7. Determine the derivative of each using the appropriate rules. Simplify final
answers. / /8
a) /(*H4*-2)2(x2+3)3 y = ̂ x4~x2, x>0
re-
8. Determine the equation of the tangent line to the curve y - at the
point (-1.-2).
c
9. An athletic-equipment supplier experiences weekly $costs ofC(x) = |x3 + 40x + 700 in producing x baseball gloves per week.
The marginal cost function is given by C"(x). Find the production level x at
which the marginal cost is $76 per glove.
= TCZVH(i
H * CT*
NAME :C: THINKING
1 0. Do the functions y = ̂ and y = x3 ever have the same tangent slope?If so when? Provide a supporting mathematical argument.
t(?r •X
TW .VI W lJ
cvuru
ycte4- 0^
W (X*or
11. Determine the value of a, given that the line ax-4y + 21 = 0 is tangent to
the graph of y--y at x = -2x
/5
\i = (X.