ES 111 Mathematical Methods in the Earth Sciences Problem Set 1 - Due Fri 30th Sept ’16 Warmup (NPC) 1a) Prove that θ cos 2 ! = 1 + cos θ 2 !1/2 [2] 1b) Use the definition of differentiation to show that if f (x) = cos x then f 0 (x) = − sin x [2]. 1c) If f (x) = xex then write down the expression for f 0 (x) [1]. 1d) Find the following indefinite integral: 1e) Find the following definite integral: R xex dx [2] R ∞ −x e dx 0 [1] 1f ) Write down the first three terms in the Maclaurin expansion for (1 − x)n [1] [9 total] Longer Questions 2 Consider a sloping roadcut (see figure). a) Write down an expression for the true thickness of the formation, d, in terms of the apparent thickness l, the roadcut dip (φ) and the bed dip (θ) [2] b) In class we showed that for a vertical roadcut, d = l cos θ. Explain how to reconcile this answer with your answer to a). [2] [4 total] 3 Consider a tin can of radius R and height f R (see figure). 1 a) Write down an expression for the total surface area A of material used to make the can [2]. b) Also write down the volume V enclosed by the can [1]. c) Using your answers to a) and b), write down an expression for A in terms of V and f only [2] d) Using differentiation, find the value of f which minimizes the surface area used to enclose a given volume. Don’t forget to demonstrate that it is a minimum, and not a maximum. [4] [9 total] 4 Here we’ll examine how we can use series expansions to make our lives easier. a) Write down the first two terms in a Maclaurin series expansion of ex . [1] The density in the Earth’s core ρ(r) varies with distance from the centre r according to ρ(r) = ρ0 e−r 2 /L2 (1) where ρ0 is the density at the centre and L is some lengthscale. b) For Earth, L is large compared to the radius of the core, so we can approximate equation (1) for ρ(r) by using a Maclaurin series expansion. Write down the first two terms in the expansion. [1] The mass of the Earth’s core M is given by M= Z R 4πr2 ρ(r)dr (2) 0 which is a tough integral if you use equation (1) for ρ(r). But if you use a series expansion it becomes much easier! c) Substitute your answer to b) into equation (2) and hence find an expression for the mass of the Earth’s core in terms of ρ0 , R and L. [4] d) If L → ∞, what does your solution reduce to? Explain why this behaviour makes sense. [3] [9 total] 2
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