2005calctest_answers

I think i found some mistakes in the answers for the review sheet Mr A gave us, so here goes.

6b) math V_x = \pi \int\limits_1^3 {\left( {4 - x} \right)^2 - \left( \right)^2 \cdot dx} math

math V_x = \pi \int\limits_1^3 {16 - 8x + x^2 - {9 \over x} \cdot dx} math

math V_x = \pi \left[ {16x - 4x^2 + {1 \over 3}x^3 + {9 \over x}} \right]_1^3 = {{8\pi } \over 3} math

15i) math {d \over {dx}}\left( {\cos ^2 x} \right) = - 2\sin x\cos x math

math \int {{{\sin 2x} \over {\cos ^2 x + 9}} \cdot dx = - } \int \cdot dx math

The expression is in the form math \int \cdot dx math so it is a natural logarithm:

math - \int \cdot dx = - \ln \left| {\cos ^2 x + 9} \right| + c = \ln \left|  \right| math

Since there is no solution for the test from 2005, lets share answers! Add your answers (and solutions if you have time) below! If you got the same answer as someone else, add a tag [confirmed: name] after. Once it's been confirmed by 2 ppl, i think we can stop with the confirm tags.

If you don't know how to do a question, just ask whoever put the answer up! :D

1) i) math 4\ln \left( {2 - x} \right) + {4 \over {2 - x}} - x + c math

1) ii) math x\arcsin x + \sqrt {1 - x^2 } + c math

1) iii) a) math {1 \over 2}\ln \left| {x^2 + 4} \right| + 2\arctan \left( \right) + c math

1) iii) b) math \ln \left| {\tan x + 1} \right| + c math

2) a) divide both sides by (T-22), integrate, get rid of natural logarithm, rearrange.

2) b) i) A = 78 ; k = -0.0323 ii) 45.4 minutes

well, im off to sleep, but here r some hints/answers for ppl who are still laboring over the test:

4) b) use first deriv = 0 for max point, and second deriv = 0 for pt of inflection

4) c) u = x, v' = cosx should give u the right answer

4) d) just chuck numbers into what you found in 4c

5) a) differentiate, plug x = 0 into first deriv. i got y = 1/8x dunno if that's right

5) b) top is variation of first deriv of bottom. gives u a natural log. answer is 10 ln (23/13)

5) c) trick is to divide top and bottom by (cosx)^2. gives u derivative on top function on bottom. natural log.

6) a) product rule. just differentiate.

6) b) u can change (cosx)^2 to (cos 2x + 1) / 2 by the double angle idneity. split fraction and manipulate it to fit it to 6a). then just plug and chuck numbers in.

good luck tmr guys! aki