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Tutor profile: Derrick B.

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Derrick B.
Mathematics tutor with 20 years of experience and success stories
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Questions

Subject: Calculus

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Question:

Use integration by parts to solve the indefinite integral $$\int \ln(x) dx$$.

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Derrick B.
Answer:

The formula for integration by parts is $$\int (u) dv=uv- \int (v) du$$. We start by observing our original integrand of $$\ln(x) dx$$. We need to determine which part is $$u$$ (which we will have to differentiate to get $$du$$) and which part is $$dv$$ (which we will have to integrate to get $$v$$). Since we cannot easily integrate $$\ln(x)$$, then we should not let that be $$dv$$. Thus, we let $$u=\ln(x)$$ and $$dv=dx$$. Since $$u=\ln(x)$$, then it follows that $$du=\frac{1}{x} dx$$. Also, since $$dv=dx$$, then it follows that $$v=\int dv=\int dx=x$$. Now we are ready to use the integration by parts formula to obtain the final answer. $$\int (u) dv=uv- \int (v) du$$; thus, $$\int \ln(x) dx=[\ln(x)](x)-\int (x)(\frac{1}{x})dx$$ $$=x\ln(x)-\int1 dx$$ $$=x\ln(x)-x+c$$.

Subject: Statistics

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Question:

Manually calculate the sample standard deviation of the following sample data set. Show all steps. $$4,6,3,7,10$$

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Derrick B.
Answer:

The formula for the sample standard deviation (represented by the symbol $$s$$) of a data set is: $$s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}}$$. First, we need to find the mean of these numbers (represented by the symbol $$\bar{x}$$). Using the following mean formula, we get $$\bar{x}=\frac{\sum_{i=1}^n (x_i)}{n}=\frac{4+6+3+7+10}{5}=\frac{30}{5}=6$$. Second, we subtract the mean from each of our original data values. This is the $$(x_i-\bar{x})$$ part of the sample standard deviation formula. We get: $$4-6=-2,$$ $$6-6=0,$$ $$3-6=-3,$$ $$7-6=1,$$ and $$10-6=4$$. Third, we square these previous answers to obtain the $$(x_i-\bar{x})^{2}$$ part of the formula. We now have: $$(-2)^{2}=4,$$ $$(0)^{2}=0,$$ $$(-3)^{2}=9,$$ $$(1)^{2}=1,$$ and $$(4)^{2}=16$$. The $$\sum_{i=1}^n (x_i - \bar{x})^2$$ part of the formula now requires us to add the numbers we obtained in the previous section. So we now have $$4+0+9+1+16=30$$ as the numerator of the formula. We now have: $$s = \sqrt{\frac{30}{n-1}}$$. The $$n$$ used in the sample standard deviation formula is the same as the $$n$$ used in the mean formula from our first step above, and it represents the total number of values in the original data set. We have $$5$$ values in this example's data set, so we can plug $$n=5$$ into the formula and simplify to obtain our final answer. $$s = \sqrt{\frac{30}{(5)-1}}=\sqrt{\frac{30}{4}}=\sqrt{7.5}=2.738612788...$$.

Subject: Algebra

TutorMe
Question:

Solve for $$x$$ by factoring: $$3x^{2}-13x-10=0$$

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Derrick B.
Answer:

This is a quadratic equation. We know this because the equation is of the form $$ax^{2}+bx+c=0$$. If we compare this general form to our problem, we see that $$a=3,$$ $$b=-13,$$ and $$c=-10$$. To factor this, we begin by looking for a pair of "magic numbers" that, when multiplied, equal $$ac$$ (which is $$(3)(-10) = -30$$) and, when added, equal $$b$$ (which is $$-13$$). If these magic numbers exist, then this quadratic expression can be factored (and thus, the equation can be solved by factoring), and there will exist only ONE pair of magic numbers that work for both conditions. If no pair of magic numbers exist, then the quadratic equation still has solutions, but cannot be solved by factoring. First, let's think about all the pairs of numbers that multiply to equal $$-30$$. We have: $$1 * -30$$ $$-1 * 30$$ $$2 * -15$$ $$-2 * 15$$ $$3 * -10$$ $$-3 * 10$$ $$5 * -6$$ $$-5 * 6$$ When we multiply each pair of numbers, we get $$-30$$. However, only one pair also adds to equal $$b=-13...$$ and that is $$2$$ and $$-15$$. So $$2$$ and $$-15$$ are our "magic numbers" for this problem. So what exactly do we DO with the magic numbers? I'm glad you asked! First we rewrite our original quadratic equation: $$3x^{2}-13x-10=0$$ Then we split the middle term ($$-13x$$) into two terms, where each term will now have a magic number as its coefficient. So for this example, $$-13x$$ now becomes $$2x - 15x$$. Thus, our equation now looks like $$3x^{2}+2x - 15x-10=0$$. Now we can factor the left side by grouping. The first two terms have a common factor of $$x$$, and the last two terms have a common factor of $$-5$$. Factoring those gives us $$x(3x+2)-5(3x+2)=0$$. Notice that we have a new common factor, because now the $$(3x+2)$$ is present in each term on the left side of the equation. So now let's factor that out to get: $$(3x+2)(x-5)=0$$. We have now factored the quadratic expression on the left side of the equation. So we are now saying that the first factor of $$(3x+2)$$ times the second factor of $$(x-5)$$ equals zero. How can that happen? How can we multiply two things and get zero as a final answer? The only way this can happen is if the first factor equals zero, or if the second factor equals zero. Thus, we set each of our factors equal to zero, and solve the remaining equations for $$x$$. This is called the "zero product property". First, we have $$3x+2=0$$. Subtract 2 from both sides to get $$3x = -2$$. Finally, divide both sides by 3 to get $$x=-\frac{2}{3}$$. This is our first solution. Next, we have $$x-5=0$$. Add 5 to both sides to get $$x=5$$. This is our second solution. We have now solved the equation. Since there are two solutions, the solution set can be written as $$\{-\frac{2}{3}, 5\}$$.

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