1. Short Answer Problems a) State the precise mathematical definition of lim f(x) = -0 b) State the Fundamental Theorem of Calculus. c) Find the exact value of tan(sec-1(4).

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17 Apr 2022

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1. (a) State the precise mathematical definition of lim f(0) = -0. (b) If B(h) is the barometric pressure in kiloPascals at height h in kilometers. What is the physical interpretation of B'(2) = 1? (c) Write the definition of tanh c and state its domain and codomain. (d) Find the exact value of log (2) – log (5) + log2 (20). (e) Use the Intermediate Value Theorem to prove that the function f(c) = r4 - 2r3 – 3.r+2 has at least one real root.

[1] 1. (a) Solve 2 – 31 < 7. Solution: We have – 7<-3 < 7 which gives -4 0 on (-0, 1] U [2,). Next, we need to see where Vr2 – 3r +2 -1 > 0. If r2 - 3.c + 2 – 1 > 0, then V.22 – 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al <, then f(c) >

5. (a) Carefully define what it means for a functionſ to be differentiable at a number a [3 points] (x + 1)3/2 + 2.0 -1 7 (b) Use your answer in part (a) to show that lim (A solution using l'Hopital's rule (if you know that) will earn 0 points) [4 points) (c) Carefully state the Fundamental Theorem of Calculus. [4 points)

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1. Short Answer Problems a) State the precise mathematical definition of lim f(x) = -0 b) State the Fundamental Theorem of Calculus. c) Find the exact value of tan(sec-1(4).

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