Take your tests: are you expert in Math?

1) What is the lowest value that the derivative of x^3-x^2+3x+1?

Answer:

\frac{d}{dx} (x^3 - x^2 + 3x + 1)
= 3x^2 - 2x + 3
= 3(x^2 - 2x/3 + 1)
= 3 [(x - 1/3)^2 + 8/9]
The lowest value of the derivative will be when x – 1/3 = 0 and its value is 3 * (8/9) = 8/3.

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?

Answer

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

\sigma(1) = 50 —> 68%
\sigma (2) = 100 —> 95%
\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?

Answer

The situation is Binomial with n = 800 and unknown p value. As an estimate of p you take 600/800 = 0.75. For a Binomial distribution with parameters n, p the mean is n*p and the standard deviation is \sqrt(n*p*(1 - p)).
\sqrt(800*0.75*0.25) = \sqrt(150) = 12.25 approx in this case.

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?

Answer

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
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 y=x^2 when x=2?

Answer

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 - y_1 = m (x - x_1)
y = 4(x-2) + 4
y = 4x – 4

6) f(x) = \frac{(\cos(x)+1)}{x-\pi}.What is the limit of f(x) as x approaches \pi?

Answer
When x = \pi, 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)]/[1]

Plug in x = \pi:
lim(x–>0)[-sin(x)]/[1] = (-sin pi)/1 = 0

Thus, lim_(x to 0)f(x) = 0

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