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Michael S.

FIU Senior, Math Major

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Trigonometry

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Question:

Show that tan(x)sin(x) + cos(x) = sec(x)

Michael S.

Answer:

We know that tan(x) = $$\frac{sin(x)}{cos(x)}$$ and that $$sin^2(x) + cos^2(x) =1$$ We'll make the left hand side of the equation into the right hand side. There are other ways to show equivalence, but this is the method I will use. tan(x)sin(x) + cos(x) = $$\frac{sin(x)}{cos(x)} * sin(x) + cos(x)$$ = $$\frac{sin^2(x)}{cos(x)} +cos(x)$$ = $$\frac{sin^2(x)}{cos(x)} + \frac{cos^2(x)}{cos(x)}$$ = $$\frac{sin^2(x)+cos^2(x)}{cos(x)} = \frac{1}{cos(x)} = sec(x)$$

Calculus

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Question:

Compute the indefinite integral $$ \int xln(x) \mathrm{d}x $$

Michael S.

Answer:

For this integral, we need to use integration by parts: $$ \int u \mathrm{d}v = uv- \int v\mathrm{d}u $$ In this case, u = ln(x) and dv= x dx So we take the derivative of u and integrate dv. du= $$ \frac{1}{x} $$ and v = $$ \frac{x^2}{2} $$ We have $$\frac{1}{2} x^2 ln(x) - \frac{1}{2}\int x\mathrm{d}x$$ The integral of x is $$\frac{x^2}{2}$$, so we are left with $$ \frac{1}{2} x^2 ln(x) - \frac{x^2}{4} +C$$

Algebra

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Question:

Write the equation of the line that passes through the points (0,5) and (-3, -1), in slope-intercept form.

Michael S.

Answer:

The slope-intercept form equation of a line is y=mx+b, where m is the slope of the line and b is the y-intercept, or where the line passes through the y-axis. We have to find the slope. The slope of a line characterizes its direction and how it angles away from the horizontal. It can also be thought of as how steep a line is; its rate of change. We are provided with two points that the line passes through, (0,5) and (-3,-1). And the way we calculate slope is the difference of the y-coordinates divided by the difference of the x-coordinates. So we have the slope as: (5-(-1)) / (0-(-3)) = (5+1) / (0+3) = 6/3 = 2 Now that we have the slope, the next thing we need to find out is the y-intercept of the line. This is where the line passes through the y-axis. At every point on the y-axis, the x-coordinate is zero. So, this was given in the question. The line passes through the point (0,5), so the y-intercept is 5. All that is left is to put it all together. y=mx+b m=2 b=5 y and x are going to change and end up giving us the points that lie on this line. y = 2x +5

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