Answer
A) 1/5, Larger
B) 1/3, Smaller
C) 1/4.1, Larger
D) $\frac{\sqrt{4+h}-2}{(h)}$
E) 1/4, By inputting numbers close to 0, negative and positive, for h
Work Step by Step
A) First find the slope between the two points:
$\frac{3-2}{9-4}$ -> $\frac{1}{5}$
Then compare this to a point further to the right to see the trend
$\frac{4-2}{16-4}$ -> $\frac{1}{6}$
as we approach values closer to $(4,2)$ from the right, we get smaller values, therefore 1/5 must be larger as it is on the right
B) First find the slope between the two points:
$\frac{2-1}{4-1}$ -> $\frac{1}{3}$
Then compare this to a point further to the left to see the trend
$\frac{2-.64}{4-.8}$ -> $\frac{1.36}{3.2}$ -> $\frac{1}{2.3529}$
as we approach values closer to $(4,2)$ from the left, we get larger values, therefore 1/3 must be smaller as it is on the left
C) 1/4.1, See reasoning used in A
D) for this question we plug in variables from the slope formula as if they were real values, we know that:
$\frac{y_{1}-y_{2}}{x_1-x_{2}}=m$
given our two points $(4,2),(4+h,f(4+h))$ we merely have to plug them in, yielding:
$\frac{2-(f(4+h))}{4 - (4+h)}$ -> $\frac{2-\sqrt(4+h)}{-h}$
E) using the previous equation, we can approximate the slope of a point by using very small values from both sides. using .0000001 as h, we get a slope of, -0.249999998481. using -.0000001 as h, we get a slope of 0.250000000701. From this we can tell that as we get closer to using the point its self from both sides, the answer approaches 1/4 or .25
