Post 1/15/2010..#22

More AP practice questions work for you??? Okay, that is what I shall do then.

EXAMPLE 1:Given the equation y = (x-3)/(2-5x). Find dy/dx.

Quotient Rule (you could technically do product, but I prefer things set in stone (plug into a formula nd all))

dy/dx = ((2-5x)(1) -(-5)(x-3))/(2-5x)^2
= (2-5x + 5x -15)/(2-5x)^2
= - 13/(2-5x)^2

What I did: I took the derivative of the function using the quotient rule**be sure not to mix up NEGATIVES!!


EXAMPLE 2: What is the maximum value for the following: f(x) = xe^-x

Take derivative using product rule (multiplying two things).

x(-e^-x) + e^-x(1)remember that the der. e is e^w/e times the der. of the exponent

Simplify:
-xe^-x + e^-x

You can factor out an e^-x

(-x+1)e^-x

To find the critical values (possible maxima) set (-x+1) equal to zero and solve for x.

This yields x = 1. Now plug in that one to e^-x.

This then gives you 1/e====>your maximum value.


EXAMPLE 3: The table shows the speed of an object in feet per second, during a 3 second period.

time(sec)-----0--1--2--3
speed(ft/sec)-30-22-12-0

Estimate the distance traveled using the trapezoid method.

All you have to do for this particular problem is find delta x then plug into the formula: delta x /2 [f(first one) +2f(next)...f(last one)]

b-a/n = delta x

3-0/3 = 1

1/2[30 + 2(22) + 2(12) + 0]

1/2[20 + 44 + 24]
=49

EXAMPLE 4: Which best describes the behavior of the function y = arctan(1/lnx) at x = t?

A. It has a jump discontinuity
B. It has an infinite discontinuity.
C. It has a removable discontinuity
D. It is both continuous and differentiable
E. It is continuous but not differentiable.

Now keep in mind that this is in the calculator portion...so if they give you a function and you have your calculuator...GRAPH it! All the question is asking you is what's happening at x = 1???

Well when looking at the graph you can clearly see that it is not continuous....(*whispers*there's a jump). So you rule out D and E. It's not a removable because there's no hole...but it clearly jumps!!! SO the answer is A!!!!!


~MalPal