# Tutor profile: David U.

## Questions

### Subject: Calculus

Find the derivate (y') for the following expression 4x^2y^7 -2x = x^5 + 4y^3

Step 1 First, we just need to find the derivative of everything with respect to x and we’ll need to recall that is really y(x) and so we can to use the Chain Rule when taking the derivative of terms involving y. This also means that the first term on the left side is really a product of functions of x and hence we will need to use the Product Rule when differentiating that term. Keynote: (On simple word this the paragraph above means , each time that we derivate respect to y we put a y' next to the term, and when we derivate respect to x we put a 1) Differentiating with respect to x gives, 8xy^7 + 28 x^2y^6y' - 2= 5x^4 +12y^2 y' Step 2 The final step is just solve for y' We subtract on both sides 12y^2 y' and 8xy^7 and we got this: 28 x^2y^6y' - 12y^2 y' -2 = 5x^4 -8xy^7 We add 2 on both sides and then we take common factor y' on the left side and we have this: y'(28x^2 y^6- 12y^2) = 5x^4 - 8xy^7 +2 And we solve for y' dividing by (28x^2 y^6- 12y^2) on both sides y' = (5x^4 - 8xy^7 + 2)/ (28x^2 y^6- 12y^2) And thats our final solution.

### Subject: Statistics

In one headquarter, 52% of the voters are devoted to David as president, and 48% are devoted to Laura as president. In other headquarter, 47% of the voters are devoted to David as president, and 53% are devoted to Laura as president. Suppose a simple random sample of 100 voters are surveyed from headquarter. What is the probability that the survey will show a greater percentage David voters in the second headquarter than in the first headquarter?

For this analysis, let P1 = the proportion of David voters in the first headquarter P2 = the proportion of David voters in the second headquarter p1 = the proportion of David voters in the sample from the first headquarter p2 = the proportion of David voters in the sample from the second headquarter. The number of voters sampled from the first headquarter (n1) = 100, and the number of voters sampled from the second headquarter (n2) = 100. Step 1 We need to make sure the sample size is big enough to model differences with a normal population. Because n1P1 = 100 * 0.52 = 52, n1(1 - P1) = 100 * 0.48 = 48, n2P2 = 100 * 0.47 = 47, and n2(1 - P2) = 100 * 0.53 = 53 Are each greater than 10, the sample size is large enough to the normal approximation. Step 2 We can find the mean of the difference in sample proportions: E(p1 - p2) = P1 - P2 = 0.52 - 0.47 = 0.05. Step 3 We can find the standard deviation of the difference with this formula. σd = sqrt{ [ P1(1 - P1) / n1 ] + [ P2(1 - P2) / n2 ] } σd = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] } σd = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706 Step 4 (Last step) We can find the probability. We need to find the probability that p1 is less than p2. This is equivalent to finding the probability that p1 - p2 is less than zero. To find this probability, we can transform the random variable (p1 - p2) into a z-score. That transformation is given by this formula. z p1 - p2 = (x - μ p1 - p2 ) / σd = = (0 - 0.05)/0.0706 = -0.7082 Using a Normal Distribution Calculator or a table, we find that the probability p(z<= -0.7082)= 0.24. And thats our final answer.

### Subject: Physics

Problem : David is driving along the street at the speed (20mph) and 30 meters before reaching a traffic light David notice it becoming yellow. Ha accelerates to make the traffic light within the 2 seconds it takes for it to turn red. What would be the required speed for David in order to cross the intersection? (We can assume that the acceleration is constant and that there is no air resistance.)

Solution We have these formulas from kinematics Formulas xf= xi + vit + 1/2 a t^2 vf = vi + at Data We can assume that xi = 0 mi We convert to units SI vi = 20 mph *( 1609.34 m/ 3600 s) = 8.94 m /s xf = 30m t = 2 sec Solve for the acceleration a= 2 (xf -vi *t)/( t^2) =2 *(30 - 8.94*2)/(2^2) = 6.06 m/s^2 Finally solve for the required speed vf = 8.94 m/s + 6.06m/s^2 * 2 s = 21.06 m/s

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