61538485 - Questions and answers PDF

Preview

Summary

Questions and answers...

Description

1. Solve for x: log2 4x = 16 log2(4x) = 16 log2(22x) = 16

We simplify 4x = 22x

2x log2(2) = 16

Then, we apply the log power rule (we put 2x in the beginning)

2x = 16

We know that log2(2) = 1

x=8

So, we found our x, which is 8

2. Find the derivative of f(x) = (e3x)4 two different ways, and the derivative of g(x) = ln (x3) two different ways



f(x) = (e3x)4

First way: We multiply the exponents, and then we use the rule of an exponential derivative which is: e^u = e^u * u’ * lne So, f(x) = e12x f'(x) = e12x * 12 f’(x) = 12 e12x

Second way: We use the rule of the exponential derivative and we also use the chain rule. f'(x) = 4 (e3x)3 * 3 e3x f'(x) = 12 (e3x)4

We used the chain rule and the exponential derivative rule. We multiply 3 by 4 to get 12 and (e3x)3 by e3x to get (e3x)4

f'(x) = 12 e12x 

g(x) = ln (x3)

First way: We use the logarithmic differentiation: ln (u) = u’/u * lne g’(x) = 3x2 / x3 g'(x) = 3 / x

Second way: We use the properties of logarithms, and then we find the derivative.

g(x) = 3 ln(x) g’(x) = 3 / x 3. If you invest $1,000,000 at 6% compounded continuously, how much interest is earned in 1 hour? We know that the equation of continuous compound interest formula is A = Pert, where A = balance, P = principle, r = rate in decimal, t = time in years To find the interest earned in 1 hour, we have to convert 1 hour to a year formula. So, we have 365 days * 24 hrs = 8760 hours in a year. Therefore, our t = 1/8760 = .000114 years A = 1000000*e0.06(0.000114) = 1000006.84

(We plug the numbers in our equation A = Pert)

Then, we have to subtract A from P. So, 1000006.84 - 1000000 = $ 6.84...