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Question 1
Solve the following equation
a) 2e
2x−1 + 5e
x
2
= 0
b) 2
4x + 22x−1 > 2
[8,6]
Question 2
Let f : (−∞, 2] → R be given by
f(x) = √
2 − x
a) Show that f is injective.
b) Determine im(f).
c) Find a left inverse g : R → (−∞, 2] of f.
[7,6,5]
Question 3
Let f : Z → Z be given by
f(z) = (
2z − 5, if z ≥ 0;
z + 5, if z < 0.
a) Is f an injective function?
b) Let u ∈ Z, u ≤ 5. show that u ∈ im(f).
c) Let v ∈ Z, v > 5, Show that v ∈ im(f) if and only if v + 5 is even .
[5,6,6]
Find the eigenvalues and the eigenfunctions of each of the following
(i) 𝑦 ′′ + 𝜆𝑦 = 0, 𝑦(0) = 0, 𝑦 ′ (𝑙) = 0,
(ii) 𝑦 ′′ + 𝜆𝑦 = 0, 𝑦(0) = 0, 𝑦(𝜋) − 𝑦 ′ (𝜋) = 0,
(iii) 𝑦 ′′ − 𝜆𝑦 = 0, 𝑦 ′ (0) = 0, 𝑦 ′ (𝑙) = 0.
Expand each of the following functions in a Fourier sine series then a Fourier cosine series on the prescribed interval.
(i) 𝑓(𝑥) = 𝑒^−𝑥 ; 0 < 𝑥 < 1,
(ii) 𝑓(𝑥) = { 𝑥 0 < 𝑥 < 𝑙/2 𝑙 − 𝑥 𝑙/2 < 𝑥 < 𝑙 ; 0 < 𝑥 < 𝑙,
Find the Fourier series for the given function 𝑓 on the prescribed interval.
𝑓(𝑥) = { −1 −1 ≤ 𝑥 < 0 1 0 ≤ 𝑥 ≤ 1 ; |𝑥| ≤ 1
Solve the following I.V.P. by method of Laplace transform:
(i) 𝑦 ′′ + 𝑦 = 𝑓(𝑡), 𝑦(0) = 0, 𝑦 ′ (0) = 0 𝑓(𝑡) = { 2 0 ≤ 𝑡 ≤ 3 3𝑡 − 7 3 < 𝑡 < ∞
(ii) 𝑦 ′′ + 𝑦 = 𝑓(𝑡), 𝑦(0) = 0, 𝑦 ′ (0) = 0 𝑓(𝑡) = { 𝑡^2 0 ≤ 𝑡 ≤ 1 0 1 < 𝑡 < ∞
Let 𝐹(𝑠) = ℒ{𝑓(𝑡)}. Prove that
ℒ { 𝑑^𝑛𝑓(𝑡)/ 𝑑𝑡^𝑛 } = 𝑠^𝑛𝐹(𝑠) − 𝑠^𝑛−1𝑓(0) − ⋯ − 𝑑^𝑛−1𝑓(0) /𝑑𝑡^𝑛−1
Hint: Try induction
Can a quadratic function have a range of (-infinity, infinity)?
Determine the largest capacity of a cylindrical tank if its surface area (without lid) should be equal to S
Calculate first order partial derivatives of a function "z = x^{2}y - 4x\\sqrt{y} - 6y^{2} + 5" at the point M(2;1)
Expand f(x) = "e^{2x} + 3x^2" in a Taylor series at a = 3 to the third derivative degree
#340153 on Dec 2023