1) What is the lowest value that the derivative of ?
The lowest value of the derivative will be when x – 1/3 = 0 and its value is .
2) A woman sells an average of 300 books per month, with a standard deviation of 50. Over the next ten years, how many months will she sell more than 375 books?
In statistics, the 68-95-99.7 rule, or three-sigma rule, or empirical rule, states that for a normal distribution, nearly all values lie within 3 standard deviations of the mean.
About 68% of the values lie within 1 standard deviation of the mean (or between the mean minus 1 times the standard deviation, and the mean plus 1 times the standard deviation).
68% of the months she will sell between 250 and 350 books.
95% of the time she will sell between 200 and 400 books.
In our case we want the sigma to be 75 so for now to get a rough calculation, and to show you how to do it once you get the correct z score, we will estimate the z score of sigma = 1.5
68+((95-68)/2) = 81.5%
This means 81.5% of the time she will sell between 225 and 375 books OR 18.5% of the time she will sell more than 375 or less than 225 Since we only want more than 375 we need to divide 18.5% by 2.
So 9.25% of the time she will sell more than 375 books.
Now just multiply that by the months. 10 years x 12 months = 120 months
120 x 9.25% = 11 months which is approximately 10 months so…10 months
3) A survey was taken of 800 drivers. 600 of them admitted that they do not obey the speed limit. What is the 95% confidence interval (to the nearest 0.01) for the true proportion who don’t obey the speed limit?
For a 95% confidence interval you need
mean +/- 1.96 standard deviations
= 600 +/- 1.96*12.25 = 600 +/- 24
Confidence interval is 576 to 624 so confidence interval for proportion is 576/800 to 624/800 or 0.72 to 0.78
4) Using continuous compounding, how long (to the nearest 0.01 years) will it take $150 to grow to $200 at 2% per year?
Final Value = Principal Value (1 + Interest Rate)^(number of years)
FV = PV(1 + i)^n
n = ?
200 = 150(1 + 2%)^n
200 = 150(1 + 0.02)^n
4/3 = 1.02^n
taking log on both sides
log(4/3) = log(1.02^n)
log(4/3) = n x log(1.02)
log(4/3)/log(1.02) = n
or n = 14.527 years
to the nearest 0.01 =>
n = 14.53 years
Therefore it will take 14.53 years (to the nearest 0.01 years) for $150 to grow to $200 at 2% per year years using continuous compounding.
5) What is tangent to when x=2?
A tangent line needs two things: a point and a slope.
The slope comes from the derivative of y = x^2, which is…
dy/dx = 2x
Evaluate that at x=2 to find the slope of the tangent line as…
dy/dx = m = 2(2) = 4
So now that we know m=4, let’s find a point that the tangent line will pass through. The function value at x=2 is
y(2) = 2^2 = 4
So we need to pass through the point (2,4) with a slope of m=4.
So here’s the equation:
y = 4(x-2) + 4
y = 4x – 4
6) .What is the limit of f(x) as x approaches ?
When , the numerator and the denominator are equal to 0.
So, use L’Hopital’s rule: take the derivative of the top, and the derivative of the bottom:
lim(x–>0)[(cos(x)+1)/(x-pi)] = lim(x–>0)[-sin(x)]/
Plug in :
lim(x–>0)[-sin(x)]/ = (-sin pi)/1 = 0
Thus, lim_(x to 0)f(x) = 0