Tutor profile: Madelyn H.

Find the vertex of the parabola of the function $$y = 2x^2 - 8x + 7$$

Remember that a quadratic equation in standard form is: $(y = ax^2 + bx + c$) The equation for the x-value of the vertex of a parabola is: $(x = -\frac{b}{2a}$) For the function given, substitute the values in for $$a$$ and $$b$$: $(x = -\frac{-8}{2\cdot 2}$) Simplify. $(x = -\frac{-8}{4} = 2$) Now, plug in $$x = 2$$ into the original equation to find the $$y$$-value. $(y= 2(2)^2 - 8(2) + 7 $) $(y = 2(4) - 16 + 7 $) $(y = -1 $) So, the vertex of the parabola is $$(2,-1)$$

Find where the function $$f(x) = x^3 + 9x^2 -48x + 2$$ is neither increasing nor decreasing.

Remember that derivative gives the rate of change of the function. Therefore, when the derivative is equal to 0, the function is not changing. First, find the derivative $(f'(x) = 3x^2 + 18x - 48 $) Simplify: $(f'(x) = 3 (x^2 + 6x - 16) $) Set the derivative equal to 0. $(3(x^2 + 6x - 16) = 0 $) To solve for 0, factor the quadratic. $(3(x+8)(x-2) = 0 $) Therefore, the derivative is equal to 0 at $$x = -8$$ and $$x = 2$$ and these are the $$x$$ values where the function is neither increasing nor decreasing.

Labor economists studying the relation between education and earnings are often concerned about what they call “ability bias.” Suppose that individuals differ in ability, and that the correct specification of the regression function is one that includes ability: $(\log(earnings)_{i} = \beta_{0} + \beta_{1} \cdot educ_{i} + \beta_{2} \cdot experi_{i} + \beta_{3} \cdot experi^2_{i} + \beta_{4} \cdot ability_{i} + \epsilon_{i}$) If we estimate the regression function with ability included, will the estimated value of $$\beta_{1}$$ will be greater or less than what it was in the regression without ability?

We know that, using the formula for omitted variable bias, the omitted variable bias for a regression without ability is: $(\beta_{4} \cdot \frac{cov(ability, educ)}{var(educ)} $) In this case, the omitted variable bias is positive. Therefore, the estimated value of $$\beta_{1}$$ with ability included would be less than the estimate of $$\beta_{1}$$ where ability is not included in the regression.