MCV4U Name:UNIT 1 TEST

THE DERIVATIVE

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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

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r-2

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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

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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.

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b) Find the velocity of the train at 90 seconds. ^/

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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)

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6. Determine the points on the curvehorizontal.

- \-I

where the tangent line is

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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

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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.

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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.

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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

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\i = (X.