1. Homework: due first meeting next week.
Instructions: For the following functions, identify the following:
1. The domain of the function
2. Intercepts
a. The x-intercepts are the points where f(x) = 0
b. The y-intercept is where x = 0
3. Asymptotes, if any.
a. Vertical asymptotes – for rational functions, they correspond to the
zeroes of the denominator after expressing it in lowest terms
b. Horizontal asymptote – evaluate the limit as x approaches positive infinity
and as x approaches negative infinity
c. Oblique asymptote – if the degree of the numerator exceeds the degree
of the denominator by 1, then the oblique asymptote correspond the
quotient when long division is performed.
4. The derivative and the critical numbers. The critical numbers should be a part of
the domain of the function, and the derivative of the function is either zero or
undefined.
5. The table of signs for the derivative, and use it to find the interval(s) where the
function is increasing and decreasing
a. If f’(x)>0, f(x) is increasing
b. If f’(x)<0, f(x) is decreasing
6. All relative extrema by using the first derivative test.
a. If f’(x) changes from positive to negative, you have a relative maximum
b. If f’(x) changes from negative to positive, you have a relative minimum.
7. The second derivative and the possible POI, those are the numbers where
f’’(x)=0 or DNE, provided that the function has a tangent line on those points.
8. The table of signs of the second derivative and use it to find the interval(s)
where the function is concave up and concave down. Also identify the POI.
a. If f’’(x)>0, f(x) is concave upward
b. If f’’(x)<0, f(x) is concave downward

2. c. Whenever f’’(x) changes sign, and the tangent line is defined on the
point, then there is a POI.
9. Try to sketch the graph given all the details you got.
a. Start with the intercepts and asymptotes
b. Next, plot the points which correspond to the relative extrema and the
POI.
c. Use the details about the derivative and the second derivative to identify
the direction of the graph at different intervals.