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Max. Marks: 100

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‎APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
FIRST SEMESTER B.TECH DEGREE EXAMINATION, APRIL 2018

‎۲ Course Code: MA101
Course Name: CALCULUS

‎PARTA
‎Answer all questions, each carries 5 marks.

‎⋅ ⋅∑⋮
↧⊃⊜⇞∁↾⋯⋯⊓∣⇂↽∣∁↥∤↧⊜∩∤↿⊜⊱⊜⊓∁≘∣≺∽ ∘∙⇟∣≮−∁∘⋂⋁⊜↾≣∈≊∙∥⊱∘⋮↥⋅⋯≺∥⇂↿⊜⊰⋃⋯

‎2
‎Examine the convergence വ്‌: ( 2“

‎Find the slope of the surface 2 = x ‏ر5 + ۲۷ء‎ in the y direction at the point ) 4, 0)

‎‘Show the function f(x,y) = e*siny + ௪70059 x satisfies the Laplace’s equation

‎fax + 1) = 0
Find the directional derivative of f (x,y,z) = x3z~ yx? + 224 ? (2, -1, 1)
in the direction of 3 7- / +2k

‎Find the unit tangent vector and unit normal vector to the curve

‎r(t) = 4costi+4sintj +tkatt = >
Using double integration, evaluate the area enclosed by the lines

‎xy
‎x = 0, ‏سر‎ = 0+ 1

‎Evaluate

‎Lean

‎7 x? +y? + 2 7)dx dy dz
0

‎0
‎If F (x,y,z) = x*i —3j + 2224८ find div ह

‎Find the work done by the force field F = ‏غ پور‎ +yzj + ‏ع عدج‎ ona particle that

‎moves along the curveC: x = ty = ८२,2 = ६3, 0 > ‏ع‎ > 1
Use Green’s theorem to evaluate [ശമു , where ೦ is the circle x? + क =

‎02

‎If S is any closed surface enclosing a volume ५ and F = xi + 297 + 32% show
that | F.nds = 6
{
PART B

‎Module I
Answer any two questions, each carries 5 marks.

‎⋅ ⋅ ⋅ k ⋅
⇂⊃∁⋢⊜∏⊺∥∏⊜↜⋁↾↿⊜⋔⊜↾⋔∊∂∣↧⊜⋯∂↥⋯⋦⊜∥∁⊴∑∘∘− പട 20501८16]
8 k=1 ಸ್‌

‎convergent
‎Find the Taylor series expansion of f (x) = ನಾ about x = 1

‎൬൩ (x+1)k

‎Find the interval of convergence and radius of convergence of 32 1(-1 ட

‎Module II
Answer any two questions, each carries 5 marks.
Find the local linear approximation L to the functionf (x,y,z) = xyz at the
pointP (1,2,3). Also compare the error in approximating f by L at the point
Q (1.001, 2.002, 3.003) with the distance PQ
Locate all relative extrema and saddle points of f ( x,y) 229 - x3-y

‎Ifu = / @,२, 0೪ thatx + > 7 + ‏مل بر‎ =

‎2

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‎Duration: 3 Hours

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