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A problem set consisting of 4 parts, each with multiple questions. The problems cover various topics in trigonometry and calculus, including limits, derivatives, integrals, and vector calculus. Students are required to find limits, derivatives, integrals, sums of series, and intersections of functions, as well as evaluate definite integrals using different methods.
Typology: Exams
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PART 1: 40 Minutes
Question 1: (10) a) Find x such that 90 < x < 180 and cos(2 x ) = โ 0.8 if cos( 37 o)= 0. 8 b) What is sin(2x)?
Question 2: (10) A continuous and differentiable function is described by f ( x )= e^2 x โ 1 for โ โ< x โค 0 , f ( x )= ax^3 + bx^2 + cx + d for 0 < x < 1 , and f ( x )= ln( x )for 1 โค x <โ. Find a, b, c, d.
Question 3: (10)
a) Find x
x x (^) sin 2 limtan^3 โ 0
b) Find f ( x )and x 0 if lim 0 3 3 f ( x 0 ) z
z z =^ โฒ
โ
Question 4: (10) Show that all such points on the curve x^2 y^2 + xy = 2 where the slope of the tangent to the curve is โ 1 , have x-coordinates that satisfy the equation x^4 + x^2 โ 2 = 0.
PART 2: 40 Minutes
Question 1: (20) (Do two of these)
โ (^) dx xx
x ( 1 )^2
Question 2: (20)
dr r
r 2
3 4
, evaluate it using
a) r^2 as one of the functions in Integration by Parts b) the substitution r = 2 tan x
PART 3: 40 Minutes
Question 1:(15 Points) Find the sum of the following series, if they exist:
a) 1 โ 1 + 1 โ 1 +โ โ โ
b) โ + โ +โ โ โ 27
20 2 (^1 )(^2 )
n n n
Question 2:(10 Points) The approximation sin( x )โ x for small x is quite often used. a) Explain this using the MacLaurin Series for sin( x ). b) Is (sin( x )โ x ) always positive, or always negative? Why?
Question 3:(10 Points)
โ n = 1 3 n
nx n converge?
PART 4: 40 Minutes
Question 1: (10) a) Find the cosine of the angle of the vector (โ 1 , 2 , 2 )with the z-axis b) Find the sine of the angle between the vectors (โ 1 , 2 , 2 ) and ( 0 , 4 , 3 )
Question 2: (5) Find the cosine of the angle between the lines y = โ 2 x + 3 and y = x โ 1 using the dot product of two vectors.
Question 3: (20) Given are the lines g (^) 1 :( a , 0 , 1 )+ t (โ 1 , b , 2 )and g (^) 2 :( 1 , 1 , 3 )+ t (โ 1 , 1 , 2 ). a) For what values of a and b will g (^) 1 and g (^) 2 be the same line? b) For what values of a and b are g 1 and g (^) 2 parallel, but not the same? c) If b = 2 , for what value of a will g (^) 1 and g (^) 2 intersect? d) For a = 1 and b = 2 , find the equation of the plane that includes g (^) 1 and is parallel to g (^) 2.
