# Tutor profile: Zachary Y.

## Questions

### Subject: Calculus

Find the equation of the tangent line with equation $$f(x)=x^2+x+2$$ at the point (2,8).

Let's recall the equation of the tangent line: $$y=mx+b$$ where m is the slope and b is the y-intercept. First, we need to find the slope. To do that, we can take the derivative of the function using the power rule: $$f'(x)=2x+1$$ Now, plug in 2 in order to get the slope at that point. $$f'(2)=2(2)+1=5$$ So the slope of the tangent line is 5, which makes the equation of the line now: $$y=5x+b$$ Now to find b, we can plug in the point for x and y and solve for b: $$8=5(2)+b$$ => $$ 8=10+b$$ => $$b=-2$$ Thus, the equation of the tangent line is: $$y=5x-2$$.

### Subject: Statistics

Scores on a history test have an average of 80 with a standard deviation of 6. What is the z-score for a student who earned a 75 on the test?

Let's recall the formula: $$z=\frac{X-\mu}{\sigma} $$ where $$\mu$$ = mean and $$\sigma$$=standard deviation. Then, since we know the mean is 80, the standard deviation is 6, and X is 75, we can plug those values into the formula to get: $$z=\frac{75-80}{6} = \frac{-5}{6} = -.83$$ So, the z-score associated with a score of 75 is -.83.

### Subject: Algebra

Solve the equation: $$ 3(x - 2)^2 - 12 = 0 $$

In this equation, we want to solve for $$x$$. So, we need to get $$x$$ by itself. In order to do that, let's follow these steps: 1. Add 12 to both sides to get: $$ 3(x-2)^2 -12 +12 = 0+12$$ $$=3(x-2)^2=12$$ 2. Divide by 3. $$\frac{3(x-2)^2}{3}=\frac{12}{3}$$ $$=(x-2)^2=4$$ 3. Square root both sides to get rid of the square: $$\sqrt{(x-2)^2} = \sqrt{4}$$ $$=x-2=2$$ 4. Add 2 to both sides to get: $$x=4$$ It's always good to check your work, so plugging 4 back into the original equation gets you: $$3(4-2)^2-12=12-12=0$$ And that is the final answer, 4!

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