Mock Midterm 2012 - MA103 Mock Midterm Name Time Allowed 80 minutes Total Value 60 marks Number of Pages 6 Instructions Non-programmable non-graphing

# Mock Midterm 2012 - MA103 Mock Midterm Name Time Allowed 80...

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MA103 Mock Midterm Name: Time Allowed: 80 minutes Total Value: 60 marks Number of Pages: 6 Instructions: Non-programmable, non-graphing calculators are permitted. No other aids allowed. Check that your test paper has no missing, blank, or illegible pages. Note that test questions appear on both sides of the paper. Answer in the spaces provided. Show all your work. Insu¢ cient justi°cation will result in a loss of marks. 1. [6 marks ] Write your answer to each of the following questions in the space provided. No justi°cation is necessary. (a) lim x !1 5 ° 1 =x = (b) lim x ! 103 p x ° 103 = (c) The domain of the function f ( x ) = sin ° 1 x is (d) True or False? If a and b are any real values, then ln ( ab ) = ln a + ln b . (e) If f ( x ) = j x + 5 j , then f is not di/erentiable at x = (f) If f ( x ) = sin ( x ) , then f (103) ( x ) = 1
2.[7marks]Consider the functionf(x) =8<:ln (°x)°3x,x± °1°°x2°x°2°°x+ 1,x >°1.(a) Determine iffis continuous atx=°1.(b) Based on our answer to part (a) alone, can we make a conclusion about the di/erentiability offatx=°1?Explain your answer. 2
3.[6marks]Use the formal de°nition of a limit to provelimx!±² sin (°)sin (2°) + tan (3°). 4.[5marks]Evaluate the limit:lim°! 3
5.[6marks]Find the equations of all horizontal asymptotes of the functionf(x) =4x°12px2°1.Justify each of your answers by evaluating an appropriate limit.6.[5marks]Prove the derivative formula for the inverse cosine function:ddx³cos°1x´=°1p1°x2. 4
7.[9marks](a) Give the precise statement of the Intermediate Value Theorem (IVT).(b) Use the IVT to show that the equationex°tan°1(x) =±2has a solution in the interval(0;1).Verify that the IVT may be applied.8.[4marks]Supposegandhare functions such thatg(x) = ln (1 +h(px)),h(2) = 1andh(2) = 1.Findg(4). 5
9.[5marks]Considerf(x) =xsinx,x >.Findf(x).10.[7marks]Determine the equation of the tangent to the curvecos (x°y) =yex°±2at the point(x; y) =µ;±2. 6
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