Ver A Math 175 Section 4
Name: ________________________________
Exam 4 Spring 2012
Show all your steps, use correct mathematical notation and simplify your answers to receive credit.
1. (a) (7 pts) Find the first 4 non-zero terms of the Taylor series for f (x) = e 3x centered about a=0.
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(b) (3 pts) Write the above series in summation notation.
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2. Recall the known Maclaurin series: sin x = ∑(−1)k
k =0
x 2k+1
, x < ∞.
(2k +1)!
sin x
(a) (10 pts) Use this series to evaluate lim
.
x →0
x
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Page 1 of 5 Ver A (b) (10 pts) Write the Maclaurin series for f (x) = sin x 3 and indicate its radius of convergence.
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(c) (10 pts) Use your series in (b) to evaluate
notation.
∫ sin( x )dx as an infinite series. Leave your answer in sigma
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3. For the parametric curve x = t, y = 4 − t, 0 ≤ t ≤ 4 :
(a) (7 pts) Eliminate the parameter to find the Cartesian form of the parametric curve.
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(b) (3pts) For what values of x and y is this curve defined?
Page 2 of 5 Ver A (c) (5 pts) Sketch the curve and indicate the direction of increasing t.
4. (10 pts) Find the equation of the line tangent to the parametric curve x = 3t 2 + 1, y = 2t 3 + 1 when t=1.
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 π
 4
5. (a) (4 pts) Express the point P 2,  in Cartesian coordinates.
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(b) (6 pts) Express the Cartesian coordinates Q(−1,0) in at least two possible polar forms.
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Page 3 of 5 Ver A 6. (10 pts) Given the parametric curve r = 4 cosθ , find an expression for
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dy
in terms of θ .
dx
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7. (a) (4 pts) Draw a graph of r against θ for r = 2 + 2sin θ . Show at least one complete cycle. For full
credit, you must label your axes and indicate any intercepts.
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(b) (6 pts) Use your graph in (a) to sketch the graph of r = 2 + 2sin θ on the x-y plane. For full credit, you
must label your axes and indicate any intercepts.
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Page 4 of 5 Ver A (c) (10 pts) Find the area of the region inside the curve r = 2 + 2sin θ between θ = 0 and θ = π .
Hint: remember to use the half angle formula when integrating any sin 2 θ or cos2 θ terms.
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Bonus (5pts): Only attempt this question after you have completed all other questions.
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Identify the function represented by the power series
xk
∑ 3k .
k= 0
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Total (out of 100 points): Page 5 of 5