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Course: MATH 130, Fall 2009
School: Allan Hancock College
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Word Count: 312

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130 Math 2008d2 -- Mathematics IE Tutorial Problems--Week 4 Calculus 1. Evaluate these limits, if possible. (a) lim x2 + 2x + x3 x 4 + 4 x (b) lim x7942 x3 x - 3 x+1-2 (c) lim x3 x-3 1 3 (d) lim - 2 x3 x - 3 x - 3x 2. Calculate (sin x)/x for some small values of x, for example, x = 1, 1/2, .1, .01, .001 and x = -1, -1/2, -.1, -.01, -.001 and see what happens. Remember that all these x-values are in radians....

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130 Math 2008d2 -- Mathematics IE Tutorial Problems--Week 4 Calculus 1. Evaluate these limits, if possible. (a) lim x2 + 2x + x3 x 4 + 4 x (b) lim x7942 x3 x - 3 x+1-2 (c) lim x3 x-3 1 3 (d) lim - 2 x3 x - 3 x - 3x 2. Calculate (sin x)/x for some small values of x, for example, x = 1, 1/2, .1, .01, .001 and x = -1, -1/2, -.1, -.01, -.001 and see what happens. Remember that all these x-values are in radians. 3. Use the limit formula for derivatives to find f (x) if f (x) = 1/x2 . 4. Find 4 dy 2 3 if y = 6x7 - 5x + 4 x - 3 + - + 2 . dx x x x 5. Find an equation for the line tangent to the graph of y = x2 at (3, 9). 1 [Short answers: 19 12 , does not exist, 1/4, 1/3; should get close to 1; 42x6 -5+(2/ x)- x-3/2 + 3x-2 - 8x-3 ; y - 9 = 6(x - 3).] 1 Algebra 6. Without using a calculator find sine, cosine, tangent and cotangent of the following angles: 3 5 7 -4 11 , - , - , , 4 4 4 3 6 7. Use periodicity property trigonometric of functions sin( 2n) = sin , cos( 2n) = cos , n Z. to find the precise value of each of the following quantities: (a) sin 35 ; 3 (b) cos 35 4 ; (c) cos(- 53 ). 6 First represent the given angle in the format 2n, where lies in the interval - < . 8. Use periodicity property of trigonometric functions tan( n) = tan , cot( n) = cot , n Z to find the precise value of each of the following quantities: (a) cot 35 4 ; (b) tan(- 37 ); 2 (c) tan(- 47 ). 6 First represent the given angle in the format n, where lies in the interval - < . 9. If cos = 1 5 and 0 < < , find other five trigonometric functions for this angle. 2 10. Derive the following identity: cos tan + sin = 2 cos . tan 11. Simplify the following expressions: (a) sin x sec x-sin2 x tan x ; 2 sin x cos x 3 2 (b) sin x + sin x cos x; (c) (sin x + cos x)2 + (cos x - sin x)2 ; (d) 1+cot x csc x - 1+tan x . sec x 2
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