[10%] Find the Taylor series of the function f (x

FINAL FOR CALCULUS

Instructor: Shu-Yen Pan

No credit will be given for an answer without reasoning.

1. [10%]

(i) Find the derivative of f (x) = ^{sin x}_x . (ii) EvaluateR_π

0(sin x + cos x) dx.

2. [10%] Find the Taylor series of the function f (x) = e^1+x at x = 1. And also find the interval of convergence.

3. [10%] Sketch the graph of the function y = f (x) = cos²x on the interval [0, π] by obtaining the following information:

(i) the intervals where f is increasing and where it is decreasing (ii) the relative extrema of f

(iii) the concavity of f (iv) the inflection points of f 4. [10%]

(i) The nth term of a sequence is given by an= ²ⁿ_3n²2⁺¹−1. Determine whether the sequence converges or diverges. If the sequence converges, find its limit.

(ii) Find the sum of the geometric seriesP_∞

n=1(¹_e)ⁿ if it converges.

5. [10%] Suppose X is a normal random variable with µ = 45 and σ = 4. Find the following values by using the table appended in the exam.

(i) P (X ≤ 50) (ii) P (40 ≤ X ≤ 50)

6. [10%] Find the mean, variance and standard deviation of the random variable x associated with the probability density function f (x) = _x⁴5 over the interval [1, ∞).

7. [10%] Find the solution of the initial value problem: y⁰ = 3x²e^−y and y(0) = 1.

8. [10%] Estimate the value√³

6 by using three iterations of the Newton-Raphson method with the initial guess x0= 2 on the function f (x) = x³− 6.

9. [10%]

(i) EvaluateR₄

1 ln x dx.

(ii) Find fx(1, −2) for f (x, y) = ^x+y_x−y.

10. [10%] Find the maximum and minimum values of the function f (x, y) = e^xysubject to the constraint x²+ y²= 8.

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