AP Calculus

finds the slope of the tangent line at f=c

more specific; at a point

use when you have a function and a point

any function at any point

can be used for every problem

as x and x+h get closer, it approaches the slope of the tangent line.

Derivative: Slope of the tangent line.

Find when taking limit of the slope of the secant line

Derivative Worksheet:

2. The answer is -3 because it is the slope of the secant line. Since the equation 9-3x is linear, it has the same slope.

4.

7.

10.

review of limits terminology:

HOMEWORK: limits graded worksheet

Challenge problem due 10/14

work on derivative packet

Major quiz Friday 10/9

-All polynomial functions continues
-Trig functions continues
-Piecewise may be continues
-Rational functions may not be continues
-Continues if:
f(c) – exist
lim(x) – exist
f(c)= lim f(x)

Quiz Clarifications

Question #1

Continuity (at point c)

Cusp

a cusp is a singular point of the curve.

(http://img.photobucket.com/albums/v48/punkdbaby/limits5.jpg)

NO DERIVATIVE AT A CUSP

Step Discontinuity

(http://www.mathwords.com/s/s_assets/s157.gif)

Removable Discontinuity

Asymptote

NO LIMIT

(http://jwilson.coe.uga.edu/EMT668/EMAT6680.Folders/Barron/Write-ups/Assignment%201/Figure4.gif)

Homework for next class

pg 58-59

23, 27, 31, 37, 41, 49, 55, 57, 65, 67, 70

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We spent the first part of class talking about the applications of derivatives. It breaks into two different concepts 1. The area under a curve or area between curves and 2. Average volume. We had already gone over the area under curves, but we covered between curves.

With the two functions, find the points of intersection to find the height of the segment that connects them. You then take the definite integral of the upper curve minus the lower. If they switch, you add the second part on to the first still using upper minus lower. Note: if one of your equations is negative, they will just add. E.g. 2- -4=6.

We then worked on the average value.

If given a velocity curve finding the definite integral of v(t)dt from points 1 to 5 gives displacement, and the displacement divided by the time interval gives average velocity. The average is ALWAYS the average value of f(x), or in terms of velocity, average turns into the average velocity.

We then spent a little bit of time reviewing the homework with led to the definite integral of 1/x is the same as the natural log.

We did problem 63 exactly.

We also noted that calc only deals with real numbers.

The homework is the Riemann Sum sheet and the 18 problems in the packet that are due Friday.

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Today we started out by discussing test taking techniques:

-Always show exactly how you found an extreme (maxima or minima) by physically setting f’(x)=0

We also began work with slope fields.  Slope fields help to determine what the family of curves will look like if the anti-derivative of a differential function is graphed. (f’(x)=2x is the differential equation of f(x)=x^2+c)

-The tangent lines represent the slope at the same point on the original function (the slope of the tangent line at a point is equal to the slope of the original function at that same point)

We also received a packet on slope fields that has problems for which we have to determine and graph the slope field.

Homework:

Do the packets and other handouts