1.Find limx→0 (1 − cos x)/x^2 given that limx→0 sin x/x= 1. (b) Use the result in part (a) to derive the approximation cos x ≈ 1 −x^2/2 as x approaches 0. (c) Use the result in part (b) to find the approximate value of cos(0.2)
1.Find limx→0 (1 − cos x)/x^2
given that limx→0 sin x/x= 1.
(b) Use the result in part (a) to derive the approximation cos x ≈ 1 −x^2/2
as x approaches 0.
(c) Use the result in part (b) to find the approximate value of cos(0.2).
2Given the function f(x) = a^x
for some real constant a.
i) Use the properties of natural logarithm and exponential functions to find
f'(x) = axM(a) and write M(a) as a natural logarithm.
ii) Use the definition of derivative to show that f'(x) = axL(a) and write an
expression for L(a) as a limit.
iii) Prove that M(a) is the limit of L(a).
(b) [5 marks] Determine whether the Mean Value Theorem (MVT) can be applied
to the function f(x) = |2x − 1| on the closed interval [−2, 4]. If the MVT can be
applied, find value(s) c ∈ (−2, 4) such that f'(c) is parallel to the chord line joining (−2, f(−2)) and (4, f(4)). If the MVT cannot be applied, give a mathematical
argument to justify your claim.
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