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When should I use the preterite tense and when should I use the imperfect tense?

The preterite and imperfect tenses are both past tenses, which are used in different situations. The preterite tends to describe things that started and finished in the past. The preterite tense indicates co...
CS
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a) Let u=(2,3,-1) and w=(3,-1,p). Given that u is perpendicular to w, find the value of p. b)Let v=(1,q,5). Given that modulus v = sqrt(42), find the possible values of q.

a) u is perpendicular to w , so u • w =0. This means that 2 3 + 3 (-1) + (-1)*p = 0, so 6-3-p=0, hence p=3. b) The magnitude of v = sqrt(1 2 +q 2 +5 2 ) = sqrt(42). Thus, 1+q 2 +25 = 42, q 2 =16, so q is eit...
CR
Answered by Cristiana R. • Maths tutor
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What is the difference between determinism and reductionism?

Reductionism is when complex behaviour is reduced to just one or two factors. For example, Behaviourism reduces complex learning down to just stimulus-response (e.g. classical conditioning), and the Biologic...
RW
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a) Given f(x) = ln(x), use the Mean Value Theorem to show that for 0 <a <b, (b-a)/b < ln(b/a) < (b-a)/a. b)Hence show that ln(1.2) lies between 1/m and 1/n, with m, n consecutive integers to be determined.

a) f(x)=ln(x), so f'(x)=1/x. By MVT, f'(c) = (f(b)-f(a))/(b-a) = (ln b - ln a)/(b-a) = ln(b/a)/(b-a), where c lies between a and b. Now, since 1/x is a decreasing function, and a &lt; c &lt; b, we get 1/b &l...
CR
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Explain Aquinas’ and Copleston’s Cosmological Argument

Finally, the third way of Aquinas’ five ways is the proof of God’s existence from contingency, which argues that God is a necessary being. Within this argument, Aquinas refers to contingent and necessary bei...
CF
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