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Question : A resistor and inductor are connected in series to a battery. The current in the circuit : 2151699

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the first three nonzero terms of the Maclaurin expansion of the given function.

1) f(x) = (1/1 + x)

A) 1 + x + x^{2} + ...

B) -x + x^{2} - x^{3} + ...

C) 1 - x + x^{2} - ...

D) 1 - x + x^{3} - ...

2) f(x) = (1/(1 + x)^{3})

A) 1 + 3x + 6x^{3} + ...

B) 1 - 3x + 6x^{2} - ...

C) 1 - 3x + 6x^{3} - ...

D) 1 + 3x + 6x^{2} + ...

3) f(x) = √(1 + x)

A) 1 + x - x^{2} + ...

B) 1 + (1/6)x - (1/9)x^{3} + ...

C) 1 + (1/6)x^{2} - (1/9)x^{4} + ...

D) 1 + (1/3)x - (1/9)x^{2} + ...

4) f(x) = (1/3√(1 + x)

A) 1 - (1/3)x + (2/9)x^{2} - ...

B) 1 + (1/6)x^{2} - (2/9)x^{4} + ...

C) 1 - (2/3)x + (2/9)x^{2} - ...

D) 1 + (1/3)x - (1/9)x^{3} + ...

5) f(x) = (-x/1 + x)

A) 2x - x^{2} + (1/2)x^{3} - ...

B) 2x + (2/3)x^{2} - (3/4)x^{3} + ...

C) -x + x^{2} - x^{3} + ...

D) x - x^{2} + x^{3} - ...

6) f(x) = e^{x} + e^{-x}

A) x + (1/6)x^{3} + (1/120)x^{5} + ...

B) 2x + (1/3)x^{3} + (1/60)x^{5} + ...

C) 2 + x^{2} + (1/12)x^{4} + ...

D) x - x^{2} + x^{3} - ...

7) f(x) = (1/2)(e^{x} + e^{-x})

A) x - x^{2} + x^{3} - ...

B) x + (1/6)x^{3} + (1/120)x^{5} + ...

C) 2x + (1/3)x^{3} + (1/60)x^{5} + ...

D) 1 + (1/2)x^{2} + (1/24)x^{4} + ...

8) f(x) = sin(4πx)

A) 4πx - 64π^{3}x^{3} + 1024π^{5}x^{5} - ...

B) 4x - (32/3)x^{3} + (128/15)x^{5} - ...

C) 4πx - (32/3)π^{3}x^{3} + (128/15)π^{5}x^{5} - ...

D) 4πx - (64/3)π^{3}x^{3} + (1024/5)π^{5}x^{5} - ...

9) f(x) = ln(1 + 2x)

A) 2x + 2x^{2} + (8/3)x^{3} - ...

B) 2x - 2x^{2} + (8/3)x^{3} - ...

C) 2x - 4x^{2} + 8x^{3} - ...

D) 2x - 2x^{2} + (4/3)x^{3} - ...

10) f(x) = sin^{2} x

A) x^{2} + (1/3)x^{4} + (2/45)x^{6} - ...

B) x^{2} + 8x^{4} + 32x^{6} - ...

C) x^{2} - 2x^{4} + (16/3)x^{6} - ...

D) x^{2} - (1/3)x^{4} + (2/45)x^{6} - ...

Find the first two nonzero terms of the Maclaurin expansion of the given function.

11) f(x) = tan2x

A) 2x - 16x^{3} + ...

B) 2x + (8/3)x^{3} + ...

C) 2x - (8/3)x^{3} + ...

D) 2x + (16/3)x^{3} + ...

12) f(x) = sec^{2}x

A) 1 - x^{2} - ...

B) 1 - 2x^{2} - ...

C) 1 + x^{2} + ...

D) 1 + 2x^{2} + ...

13) f(x) = sinx^{2}

A) x^{2} + (1/6)x^{3} + ...

B) x^{2} - (1/6)x^{3} + ...

C) x^{2} - 120x^{6} + ...

D) x^{2} - (1/6)x^{6} + ...

14) f(x) = e^{x}sinx

A) x + 2x^{2} + ...

B) x - (1/2)x^{2} + ...

C) x + x^{2} + ...

D) x - x^{2} + ...

15) f(x) = etanx

A) x + x^{2} - ...

B) 1 + x^{2} + ...

C) (1/2) + 2x + ...

D) 1 + x + ...

16) f(x) = e^{x}cosx

A) (1/2) + 2x - ...

B) x + x^{2} - ...

C) 1 + x^{2} + ...

D) 1 + x - ...

17) f(x) = x^{e}cosx

A) x - x^{3} + ...

B) x - (x^{3}/2) + ...

C) ex - e^{3}x^{3} + ...

D) ex - (ex^{3}/2) + ...

18) f(x) = tan^{2}x

A) x + (1/2)x^{2} + ...

B) x - (2/3)x^{2} - ...

C) (1/2) - 2x + ...

D) x^{2} + (2/3)x^{4} + ...

19) f(x) = x^{e-x^2}

A) x - x^{3} + ...

B) x + x^{3} + ...

C) x - (1/2)x^{2} + ...

D) x + (1/6)x^{3} + ...

20) f(x) = e^{x^2}

A) 1 + x^{2} + ...

B) 1 - x^{2} + ...

C) 1 + (1/2)x^{2} + ...

D) x - (1/2)x^{2} - ...

Solve.

21) The displacement y (in cm) of an object hung vertically from a spring and allowed to oscillate is given by the equation y = 6e^{-0.1t} sin t, where t is the time (in s). Find the first three terms of the Maclauren expansion of this function.

A) 6t + 0.6t^{2} + 0.03t^{3} + . . .

B) 6t + 1.2t^{2} - 1.94t^{3} + . . .

C) 6t + 0.6t^{2} - 0.97t^{3} + . . .

D) 6t + 0.3t^{2} - 0.98t^{3} + . . .

22) A resistor and inductor are connected in series to a battery. The current in the circuit (in A) is given by I = 1.2(1 - e^{-0.1t}), where t is the time since the circuit was closed. Find the first three terms of the Maclauren expansion of this function.

A) 1.2t + 0.006t^{2} + 0.0002t^{3} + . . .

B) 0.12t - 0.006t^{2} + 0.0002t^{3} + . . .

C) 0.12t - 0.06t^{2} + 0.02t^{3} + . . .

D) 0.12t - 0.006t^{2} + 0.0004t^{3} + . . .

23) The amount (in g) of a radioactive substance remaining after time t (in months) is given by the function A = 200e^{-0.03t}. Express A = f(t) in polynomial form by using the first three terms of the Maclauren expansion.

A) 200 - 6t + 0.09t^{2} + . . .

B) 200 - 200t + 100t^{2} + . . .

C) 200 - 6t + 0.18t^{2} + . . .

D) 200 + 6t + 0.09t^{2} + . . .

24) The monthly sales (in thousands of dollars) of a small business can be modeled by the function S = 50 + 18cos(πx/6), where x is the time in months, with x = 0 corresponding to July. Find the first three terms of the Maclauren expansion of this function.

A) 68 - (1/2)π^{2}x^{2} + (1/72)π^{4}x^{4} + . . .

B) 68 - 3πx- (1/4)π^{2}x^{2} + . . .

C) 68 - 9x^{2} + (3/4)x^{4} + . . .

D) 68 - (1/4)π^{2}x^{2} + (1/1728)π^{4}x^{4} + . . .

25) Is it possible to find a Maclaurin expansion for f(x) = (1/1 - cosx)? Explain.

A) Yes, the function has derivatives of all orders at x = 0.

B) No, the function is not defined at x = 0.

26) Is it possible to find a Maclaurin expansion for f(x) = cotx? Explain.

A) No, the function is not defined at x = 0.

B) Yes, the function has derivatives of all orders at x = 0.

27) Use the Maclaurin expansion of e^{x} to find (x→0 is under lim) (1 - e^{x}/x).

A) ∞

B) -1

C) 1

D) 0

28) Use the Maclaurin expansion of cos x to find (x→0 is under lim) (1 - cosx/x^{2}).

A) 0

B) (1/2)

C) 1

D) ∞

29) Is it possible to find a Maclaurin expansion for f(x) = x^{3/2}? Explain.

A) Yes, the function has derivatives of all orders at x = 0.

B) No, the second-order and higher derivatives for this function are not defined at x = 0.

30) Is it possible to find a Maclaurin expansion for f(x) = x^{3} + 5x? Explain.

A) No, the function does not have fourth-order or higher derivatives at x = 0.

B) Yes, a polynomial function is equal to its Maclaurin expansion and has derivatives of all orders at x = 0.

31) Find the Maclaurin expansion for the function f(x) = (3 + x)^{3}.

A) The Maclaurin expansion is not valid because it is effectively not infinite.

B) The Maclaurin expansion cannot be found because all terms past a certain point are zero.

C) 27 + 27x + 9x^{2} + x^{3}

D) 3 + 9x + 9x^{2} + x^{3}

32) Find the Maclaurin series for y = sinh 5x.

A) 1 - (25/2)x^{2} + (625/24)x^{4} - . . .

B) 1 + (25/2)x^{2} + (625/24)x^{4} + . . .

C) 5x + (125/6)x^{3} + (625/24)x^{5} + . . .

D) 5x - (125/6)x^{3} + (625/24)x^{5} - . . .

33) Find the Maclaurin series for y = cosh 6x.

A) 1 + 18x^{2} + 54x^{4} + . . .

B) 1 - 18x^{2} + 54x^{4} - . . .

C) 6x + 36x^{3} + (324/5)x^{5} + . . .

D) 6x - 36x^{3} + (324/5)x^{5} - . . .