## Midterm II Review for Math 126, Spring 2010

#### Topics we covered.

1. (last) differentiation techniques
1. the Chain Rule
2. implicit differentiation
2. applications of differentiation
1. terminology from natural and social sciences:
1. derivative of position (or displacement) is velocity, whose derivative is acceleration, whose derivative is jerk, whose derivative is snap, whose derivative is crackle, whose derivative is pop.
2. (linear) density is the derivative of (linear) mass
3. the word "marginal" in economics always means "derivative of"
4. any "rate of change" is a derivative of some underlying quantity
2. Related Rates; when in doubt, use the method:
1. clearly define all variables, with words and also (if possible) a picture; notes:
• there must be enough variables to do the next two steps
• variables are things that vary -- no need to have a variable for a quantity which does not change during the problem (e.g., in "a 2m tall man walks..." you don't need a variable for the man's height)
• but don't put numbers on the picture (or into any equations until the very end of you work) for quantities which have a particular, stated value only at one instant (e.g., in "how long is the man shadow when he is 3m from the wall" you should not put that 3 in your diagram nor into your calculus/algebra until the very end)
2. state the given information in terms of your variables
3. pose the problem's question in terms of your variables
4. find a "relationship equation" (or several) between the variables in the previous two steps; these usually come from some standard mathematics or other science, such as:
• the Pythagorean Theorem
• basic trigonometry (SOHCAHTOA)
• the Law of Cosines or Law of Sines
• forumlæ for the area or volume of some standard geometric object (a triangle, circle, rectangle, sphere, pyramid, rectangular parallelepiped)
• some basic physics such Newton's Second Law or Law of Universal Gravitation, or the relationship between position, velocity, acceleration, jerk, snap, crackle, and pop.
5. use the above three steps to solve the problem, usually by differentiating the relationship equation (use the Chain Rule a lot!) and plugging in the known information to solve for the unknown
6. restate the solution in the same terms with which the problem was posed.
You do not have to follow those steps in exactly that order or self-consciously announcing the steps as you do them. You do have to have essentially all of the above content, and to explain what the content is as you provide it. In particular, student most frequently loose points for failing to define their variables or to have a complete and coherenet explanatory narrative of their work.
3. linear approximations: approximate the value of a function at some x by using the equation of the tangent line y=f(a) + f(a)(x-a) based at some nearby point (a,f(a)).
3. max/min in the abstract:
1. definition of local (or relative) and global (or absolute) maximum, minimum, and extremum -- note the endpoint of an interval of definition of a function can never be a local extremum. [Note the plural of the word extremum is extrema).]
2. the Extreme Value Theorem: A continuous function defined on a closed interval has a global maximum and global minimum in that interval.
3. Fermat's Theorem: If a function f(x) has a local extremum at some point c where f(c) exists, then necessarily f(c)=0.
4. a critical number for a function f(x) is a value c in the domain of f for which either the derivative f(c) does not exist or, if it exists, it equals 0.
5. the Closed Interval Method to find the extrema of a function on an interval [a,b]:
1. find the critical numbers of f(x) inside the interval (a,b) by computing the derivative f(x) and seeing where it fails to exist or exists but equals 0 (Always check for both kinds of C#!) for x in the interval
2. compute the values of f at each such critical number and at the endpoints of the interval
3. pick off the largest and smallest values just computed: these are the global extrema of f on the given interval
4. a pair of "existence theorems":
1. Rolle's Theorem: If f(x) is a function which
• is continuous on some interval [a,b],
• is differentiable on (a,b), and
• satisfies f(a)=f(b),
then there exists some c in (a,b) such that f(c)=0.
2. the Mean Value Theorem [MVT]: If f(x) is a function which
• is continuous on some interval [a,b], and
• is differentiable on (a,b),
then there exists some c in (a,b) such that f(c)=(f(b)-f(a))/(b-a).
When you use either of these theorems, you must check (and explain why you believe) all of their hypotheses.
5. what first and second derivatives tell about the shape of the graph of a function:
1. the Increasing/Decreasing Test:
• If f(x)>0 on some interval, then f(x) is increasing on that interval.
• If f(x)<0 on some interval, then f(x) is decreasing on that interval.
2. the First Derivative Test:
• If f(x) changes from positive to negative at some value x=c, then f has a local maximum at c.
• If f(x) changes from negative to positive at some value x=c, then f has a local minimum at c.
• If f(x) does not change sign at x=c, then c is neither a local maximum nor local maximum for f.
3. If, on some interval, the graph of f(x) lies above all of its tangents, then we say that f(x) is concave upwards on that interval; if below, we call it concave downwards.
4. the Concavity Test:
• If f′′(x)>0 on some interval, then f(x) is concave upwards on that interval.
• If f′′(x)<0 on some interval, then f(x) is concave downwards on that interval.
5. If the concavity of a function changes at a point, we call that point an inflection point for the function.
6. the Second Derivative Test: If x=c is a critical number for a function f(x) and f′′(x) exists and is continuous near c, then:
• if f′′(c)>0, then f has a local minimum at c;
• if f′′(c)<0, then f has a local maximum at c;
• if f′′(c)=0, then c might be a local extremum for f, but it also might not.
6. limits to infinity:
1. we say the limit as x goes to ∞ of a function f(x) is the number L if we can make the values of f(x) as close to L as we like, by making x sufficiently large (similarly if we are taking the limit to -∞, except we make x sufficiently large and negative)
• if r>0 is a rational number, then the limit as x goes to ∞ of 1/xr is 0;
• if r>0 is a rational number such that xr is defined for all x, then the limit as x goes to -∞ of 1/xr is 0;
• if r>0 is a rational number, then the limit as x goes to ∞ of xr is ∞.
3. we generally compute limits to ±∞ by working with a fraction, dividing the top and bottom by the appropriate power of x (often the highest power, counting powers appropriately if they are inside roots), using the Limit Laws and the above theorem about powers until we get something with only legal arithmetic in it
4. arithmetic which we can do with infinity:
• (±c) ∞ = ± ∞, where c is a finite, positive constant
• ∞ + ∞ = ∞
• ∞ . ∞ = ∞
5. arithmetic which we should absolutely avoid with infinity:
• ∞ - ∞
• ∞/∞
• 0 . ∞
7. graphing functions:
1. a strategy (or checklist) for graphing a function f(x) from the book:
1. Domain
2. Intercepts (x and y)
3. Symmetry/periodicity
4. Asymptotes (horizontal, vertical, and slant)
• slant asymptotes occur only in rational functions where the numerator has degree exactly one more than the denominator; in this case, the SA can be found by long division of polynomials
5. Intervals of Increase and Decrease
6. Local Extrema
7. Concavity and Points of Inflection
8. Sketch the curve
Note that it is permissible in our class to estimate all of the above features based on the graph from your calculator, but, unless specifically told not to, you should always also use calculus techniques to find exact values.
2. pitfalls with calculators: mostly, it amounts to finding the right window, so it helps to use calculus to find local extrema using calculus, for example, and then to choose a window that contains them
8. optimization problems
1. a method can be very similar to the above method for Related Rates problems
1. clearly define all variables -- here, the criteria usually is to have enough variables to describe completely the problem scenario and how it can vary
2. find any given information in terms of your variables; also, find an admissible interval for the quantities you can vary, if there is one
3. state the goal, to maximize or minimize some objective or target function
4. find as many relationship equations between your variables as you can: often, there will be one or more equations which yield a constraint equation for your input (independent) variables
5. solve: often, this involves using the constraint equation(s) to eliminate all but one independent variable in your target function, so that you can then either use
• the Closed Interval Method (if there was an interval) or
• techniques from the graphing section, above, to find a global extremum even when there is no closed interval
6. state the answer in the terms in which the problem was posed.

 Jonathan Poritz (jonathan.poritz@gmail.com)