Calculus Unit 1 Test

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UNIT 1 TEST THE DERIVATIVE

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An overall comrptlhication mark will be assigned for proper use of mathematical form, termipeiogy and conventions. DERIVATIVE RULES MAY BE USED EXCEPT WHERE LIMIT THEORY IS SPECIFIED. A: KNOWLEDGE/UNDERSTANDING 1. Determine the value of each of the following limits if it exists. ,. b)

a) lim—= '

3-VX + 9 ton

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c) lim —*

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V „,-, I

r-2

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\n 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

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a) y =

4. A subway train travels from one station to the next. Its distance, in kilometres, 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: APPLICATION 5. a) Use the Product rule to determine the rate of change of f(x) = x2 (3x -1)

at x = 3 .

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b) Show that you get the same result using the limit definition of the derivative.

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c) Show another method that could be used to verify the value of /'(3)

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6. Determine the points on the curve horizontal.

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where the tangent line is -I

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7. Determine the derivative of each using the appropriate rules. Simplify final answers.

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2

a) /(*H4*-2) (x +3)

3

4

2

y = ^x ~x , x>0

re8. Determine the equation of the tangent line to the curve y -

at the

point (-1.-2).

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9. An athletic-equipment supplier experiences weekly $costs of C(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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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 = -2 x

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