Mathematical Scratchpad
Math 151 Review for final.
If the graph of
is given together with its first and second derivatives below. Label the
graphs for
and
Ans: is red, is thin and is black thick.
If the graph of is given below:
Compare in increasing order. [ Ans:
Suppose the function (graph of , is given above) represent the distance that a car travels.
When do you think the car has zero velocity between and ? [Ans: look for the point where so it is about x=4 (max of f) or x=0.8 (min of f ].
Approximate the time when the car reaches its highest speed for and . [Note: speed = ].
[Ans: We look for the slope of to be the largest, so it is the inflection point of .]
Find the tangent line equations and the at a given point for the following functions:
at [Ans: y=0.6708 x+1.565]
at [y=-3*x-4]
Find [Hint: Identify a function and a point and use the derivative at one point.]
[Ans. This limit is the same as when we pick and so it is ]
Find the following limits with explanations:
[ans. ]
[
[
[=0
[=0
[=0
Let the graph of a velocity function (x = time, y = feet/sec) be given below: Assuming the x-intercepts for the following graph is at and
What is the initial velocity (when time=0)? [Ans: Since when , is not found, so we do not have enough info to find the initial velocity.]
Explain how velocity function can be negative sometimes. [Ans. When we travel in opposite direction, the velocity is negative.]
Estimate the inflection point(s) for the distance function[Ans, they are at the local min and the local max of the velocity, so it about and ].
Estimate the maximum and minimum for the distance function[Ans, max is at and minimum at
Find the interval(s) where the acceleration is negative. [Ans. This is where the velocity has negative slopes].
If the graphs of and are given below: Then
identify the graph for and respectively. [Ans. is the one always positive and the last one is for ]
find the interval(s) where is increasing or decreasing, [ans. is always increasing]
find the maximum and minimum for in the interval [since is increasing in the minimum is at and the maximum is at
find the interval(s) where is concave upward and concave downward. [Since in , is concave in this interval;similarly, in is concave down this interval.]
Find the first derivative for the following functions:
[
[
[
[done in class]
[
[
If the derivative of a function is Then
find the interval(s) where original function is increasing and decreasing, [hint: draw
find the interval(s) where the original function is concave upward and concave downward. [hint: find and use the signs of to find the intervals where is concave upward or downward]
graph one possible function which has the derivate function **this is a typo, ignore this one]
Find if [Use implicit differentiation to find
Find if [ ].
Find [
Find the first derivative for the following functions:
[
[
Use the product or quotient rule to find
Find the followings:
[
[
If Find by hand.
Suppose the cost, in dollars, for a company to produce pairs of a new line of jeans is
Find the marginal cost function. [
Find and explain its meaning. What does it predict? [This will predict the cost for the 101st unit]
Estimate the cost of making the 99 th pair of jeans. [Use use calculator to find this value]
If
use the signs of to find the intervals where is increasing or decreasing, [hint: plot , which is shown above. We need find the zero of which is at so is increasing in and is decreasing in ]
find the relative maximum and minimum for [ has a relative minimum at
find [hint:
use the signs of to find the intervals where is concave upward or downward. [hint: after finding you need to plot