Sma2161:Mathematics Ii Question Paper
Sma2161:Mathematics Ii
Course:Bachelor Of Horticulture
Institution: Meru University Of Science And Technology question papers
Exam Year:2012
QUESTION ONE (30 MARKS)
a) Evaluate the following limits i. lim???2 ??3-??2-??-2 ??-2 (3 Marks) ii. lim???8 10??4+??3-3??2+4??+2 2??4+??2+5 (3 Marks) b) Differentiate the following function implicitly with respect to x. ??2??3 - ????2 = 10???? (4 Marks) c) By use of first principles obtain the derivative of ?? ?? = 3??3. (4 Marks) d) Evaluate the following integral by parts ??2 cos??????. (4 Marks) e) Find ???? ???? given that ?? = cos3 ???????? ?? = sin3 ??. (4 Marks) f) Differentiate the following using the quotient rule. (4 Marks)
?? =
2??3+3??2 ??4+1
g) Find the area between the curves ?? = ??2 - 4 and the x-axis from ?? = -2 to ?? = 2. (4 Marks)
QUESTION TWO (20 MARKS)
a) Integrate the following by rule indicated i. ??3 ln?????? by parts (3 Marks) ii. ??+1 3??2+6??+5 3 ???? (substitution). (3 Marks) iii. ??-11 ??+3 (??-4) ???? (partial fractions) (3 Marks)
2
b) i) Sketch the curve ?? = ??3 - 3??2 + 2?? from ?? = 0 ???? ?? = 2. (3 Marks) ii) Find the area between the curves ?? = ??3 - 3??2 + 2?? and the x-axis from ?? = 0 ???? ?? = 2. (4 Marks) iii) Find the volume of the solid generated by revolving the region between the curve ?? = ??2 + 1 and the x-axis from ?? = -1 ???? ?? = 1 through 360° about the x=axis. (4 Marks)
QUESTION THREE (20 MARKS)
a) Use logarithmic differentiation to evaluate the derivative of the following. i. ?? = ??sin ?? (5 Marks) ii. ?? = (??2+1)9(??-3)3 4 (??-5) 2 3 (??2+2??+1)-1 3 (8 Marks) b) Find the values of the constants so that the following functions are continuous on the entire real number line. (7 Marks)
?? ?? =
2 ?? = -1 ???? + ?? -1 < ?? < 3 -2 ?? = 3
QUESTION FOUR (20 MARKS)
a) Find the derivatives of the following by the rule indicated. i. ?? = ??2 - 1 ??3 + 2?? (??- 1 2 - 4??) (product rule). (4 Marks) ii. ?? = ??2+2 ??3-4?? (quotient rule). (3 Marks) iii. ?? = [(6?? + 8)3 + 10]5 (chain rule) (4 Marks) b) Find the derivatives of ?? = tanh??. (4 Marks) c) From first principles find the derivative of ?? = tan??. (5 Marks)
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