Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 4 views

    223 Ee

    Uploaded by

    Gilbert Aggrey
    assignment

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    223 Ee

    Uploaded by

    Gilbert Aggrey
    4 views2 pages
    assignment

    Original Title

    223ee

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    assignment

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    4 views2 pages

    223 Ee

    Uploaded by

    Gilbert Aggrey
    assignment

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    UNIVERSITY OF GHANA

    (All rights reserved)


    BSc/BA, FIRST SEMESTER EXAMINATIONS: 2017/2018
    DEPARTMENT OF MATHEMATICS
    MATH 223: CALCULUS II (3 credits)
    INSTRUCTION:
    ANSWER ANY 4 OUT OF THE FOLLOWING 6 QUESTIONS
    TIME ALLOWED:
     
    1
    TWO HOURS AND THIRTY MINUTES 2 hours
    2

    1. (a) Use the Mean Value Theorem to establish the following inequality

    ea (x − a) < ex − ea < ex (x − a),

    if a < x. [15 Marks]


    (b) By considering the derivative of the function f : [−1, 1] → R defined by
    2x
    f (x) = ,
    x2+1
    0
    show that f −1 exists and find (f −1 ) ( 45 ). [15 Marks]
    (c) Evaluate the following limit
     x + 1 x
    lim .
    x→∞ x+2
    [20 Marks]

    2. (a) Express 5 sinh x + cosh x in the form Aex + Be−x , where A and B are integers. [10 Marks]
    (b) Solve the equation 5 sinh x + cosh x + 5 = 0, giving your answer in the form ln a, where
    a ∈ R. [15 Marks]
    (c) Differentiate the following functions with respect to x
    Z x
    sin t
    (i) sinh x tanh x (ii) √ dt. [25 Marks]
    x t

    EXAMINERS:Miss. Lilian F. Kyei, Dr. Joseph Ansong and Page 1 of 2


    Dr. Asare-Tuah Anton
    3. (a) Use the Riemann sum to calculate the area under the curve y = −x2 + 3x + 2 between
    x = −1 and x = 2. [20 Marks]
     
    n−1
    4j 2 4 
    X  
    (b) Let R = lim  3+ · , write R as a definite integral and hence evaluate R.
    n→∞ n n
    j=0

    [15 Marks]
    (c) If xy = ex−y , prove that
    dy ln x
    = .
    dx (1 + ln x)2
    [15 Marks]

    4. (a) Evaluate the following integrals


    Z Z
    x+1 dx
    (i) √ dx (ii) (Hint: use the substitution t = tan(x/2) )
    x2 − x + 1 5 + 3 cos x
    [25 Marks]
    (b) Evaluate the following definite integrals
    Z 2
    dx
    (i) 2 2
    (Hint: Use the substitution x = 2 tan θ)
    0 (4 + x )
    Z 2
    (ii) x4 (ln x)2 dx (Hint: Use integration by parts). [25 Marks]
    1
    d
    5. (a) By writing 3 cos x + 4 sin x = λ dx (4 cos x + 5 sin x) + µ(4 cos x + 5 sin x), where λ and µ are
    constants, find the values of λ and µ and hence evaluate the integral
    Z
    3 cos x + 4 sin x
    dx.
    4 cos x + 5 sin x
    [15 marks]
    Z 1 p
    (b) Let In = xn 1 − x2 dx, n ∈ N. Show that (n + 2)In = (n − 1)In−2 , n ≥ 2. [20 Marks]
    0
    (c) If g = sin(sin x), prove that

    d2 y dy
    2
    + tan x + y cos2 x = 0.
    dx dx
    [15 Marks]

    6. (a) Evaluate the following improper integrals


    Z 3 Z ∞ ∞
    6x3
    Z
    dx
    (i) (ii) (1 + 2x)e−x dx (iii) dx.
    0 (x − 1)2/3
    0 −∞ (x4+ 1)2
    [35 Marks]
    (b) Evaluate the following integral using partial fractions

    3x2 + 2x
    Z
    dx.
    (x + 2)(x2 + 4)

    [15 Marks]

    EXAMINERS:Miss. Lilian F. Kyei, Dr. Joseph Ansong and Page 2 of 2


    Dr. Asare-Tuah Anton

    You might also like