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    Exercises MEF - 5 - 2018

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    rtchuidjangnana
    The document contains exercises on calculus topics like limits, continuity, Taylor series approximations, and partial derivatives. There are 8 exercises involving functions of one or more variables and applying calculus concepts like limits, derivatives, and Taylor approximations.

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    Exercises MEF - 5 - 2018

    Uploaded by

    rtchuidjangnana
    21 views2 pages
    The document contains exercises on calculus topics like limits, continuity, Taylor series approximations, and partial derivatives. There are 8 exercises involving functions of one or more variables and applying calculus concepts like limits, derivatives, and Taylor approximations.

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    Travaux pratiques 5 math

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    Exercises MEF_5_2018

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    The document contains exercises on calculus topics like limits, continuity, Taylor series approximations, and partial derivatives. There are 8 exercises involving functions of one or more variables and applying calculus concepts like limits, derivatives, and Taylor approximations.

    Copyright:

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

    Exercises MEF - 5 - 2018

    Uploaded by

    rtchuidjangnana
    The document contains exercises on calculus topics like limits, continuity, Taylor series approximations, and partial derivatives. There are 8 exercises involving functions of one or more variables and applying calculus concepts like limits, derivatives, and Taylor approximations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
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    Exercise Session 5, 23 October 2018

    Mathematics for Economics and Finance


    Prof: Norman Schürhoff
    TAs: Jakub Hajda, Jimmy Lerch

    Exercise 1
    Calculate the limit of the following functions for x → ∞.

    (a) f (x) = x ln 1 + xr .
    

    x
    (b) g (x) = 1 + xr .

    Exercise 2
    a) Let f, g be two continuous functions on [a, b] ⊂ R and let x0 ∈ (a, b) with f (x0 ) = g(x0 ). We then define
    (
    f (x) if x ∈ [a, x0 ),
    h(x) =
    g(x) if x ∈ [x0 , b].

    Verify that h is continuous on [a, b].


    b) Verify that h is continuous on R, where h is given by

    1 if x ∈ (−∞, −1),

    h(x) = x2 if x ∈ [−1, 2),

    x + 2 if x ∈ [2, ∞).

    Exercise 3
    Given is a function sin(x)

    (a) Use Taylor series representation to represent sin(x) as an infinite sum.

    (b) Use a second order Taylor approximation to approximate sin(0.1) around 0. Compute the approximation
    error.

    Exercise 4
    Consider the function f (x) = ln(x2 + x − 1). Compute the exact value of f at x = 1.1 and compare it to the
    third-order Taylor approximation in the neighborhood of x = 1. What is the order of the approximation error?

    1
    Exercise 5
    Given is a variable x ∈ R which takes the value xt at time t (e.g. GDP). The expression
    xt − xt−1
    γ=
    xt−1
    is the growth rate at time t.
    (a) Show that the growth rate γ can be approximated by 4ln(xt ) = ln(xt ) − ln(xt−1 ). (Hint: Use first order
    Taylor approximation around xt−1 )
    (b) Find the expression for the approximation error of the first order Taylor approximation.
    (c) Argue using the results from b) why the first order Taylor approximation of γ is better if 4xt → 0.

    Exercise 6
    The price of a zero-coupon bond with face value F and maturity T as a function of the interest rate r is given by
    F
    P (r) = .
    (1 + r)T
    We are interested in examining the behavior of the bond price when the interest rate changes from its initial value
    r0 to r1 = r0 + ∆r.
    (a) Compute the first- and second-order derivatives of P (r). How would you interpret these derivatives (that
    is, consider their signs and effects)?
    (b) State the Taylor’s theorem for P (r1 ). Compute the first- and second-order approximation of P (r) at r1 .
    What happens to the second-order term in the Taylor’s expansion when ∆r is close to 0?
    (c) Prove that the actual bond price P (r1 ) is always larger than its first-oder approximation when ∆r is greater
    than 0. What does your result imply for approximating bond prices using Taylor’s expansion? Hint: consider
    the sign of the second-order term.

    Exercise 7
    Assume
    1
    (x2 + y 2 ) sin x2 +y x2 + y 2 =
    
    2, 6 0,
    f (x, y) =
    0, x2 + y 2 = 0,
    (a) Show that all first-order partials exist, and check whether they are continuous.
    (b) Check whether the above function is differentiable at (0, 0).

    Exercise 8
    Let f : R2 → R be
    √ xy
    (
    if (x, y) 6= (0, 0),
    f (x, y) = x +y 2
    2
    .
    0 if (x, y) = (0, 0).
    (a) Is f a continuous function?
    (b) Is f a differentiable function?

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