created by Acces for use by Park Tudor Upper School only.
created by Acces for use by Park Tudor Upper School only.
created by Acces for use by Park Tudor Upper School only.
Calculus
Chapter 4 Lesson 5 Worksheet
Names:_
1. Given the graph of f(x) identify the following information.
a) identify where f (x) is increasing
b) identify the critical points
c) identify the relative extrema
and give their type
d) the absolute maximum
Calculus
Chapter 4 Lesson 1
In chapter 4 we will examine the behavior of functions using calculus:
*If f(x) is defined at the point c then
*If f (c) < 0 then the function is DECREASING at c
*If f (c) > 0 then the function is INCREASING at c
*If f (c) = 0
AP Calculus Test Chapter 3A Derivatives
Oct. 9, 2015
NO CALCULATORS !
Multiple Choice: Select the Best Answer
Name:_
Free Response CLEARLY show ALL steps that lead to your answer.
10. Use the Limit Definition of a derivative to find the derivative of f (
AP Calculus Review Worksheet Chapter 1: PreCalculus Review
Multiple Choice: Select the best answer Name:
NO calculators for this worksheet
Which of the following defines a mction f for which f(x) = f (x) ?
(A) f(x) = x2 (B) f (x) = $in (C) f (x) = 008x
(D
Calculus Chapter 3
Lesson 5
Section 3.3
We have developed many useful short cut rules to find derivatives.
Ex: find the derivatives of the following functions:
2
f ( x) 3x 5 2 x 4 5x 3 7(e x )
3
g ( x) 3x 4 (e x )
We need to develop a rule for the multipl
Calculus Chapter 2
Lesson 5
Evaluate the limit:
lim x0 sin 2 x + 4 cos x + 3 x + 5
lim x0
cot x
csc x
lim x x 3 tan x
2
Why are the rules that we covered in lesson 3 & 4 not helpful in the second and third problems?
How can we find a solution?
If you are
Calculus Chapter 3 Lesson 16
We know the derivative of:
y ex
and even
Now find the derivative of
Hint: if
Rule
y ln x
then
y e4 x
2
2x
y ln x
ey
x
1
Dx ln x
x
If we know the derivative y ln x what about the derivative of y log5 x
Hint: i) how do you find
Calculus
Chapter 3
Lesson 7
Section 3.6: Derivatives of Trig functions
Given the graph of y=sin(x),
graph y
Can we find an equation algebriacally ?
lim h
lim h
0
sin( x h) sin( x)
h
lim h
sin x(cos h 1) cos x sin h
0
h
0
sin( x) cos(h) cos( x) sin(h) sin(
Calculus
Chapter 3
Lesson 2
(section 3.1 Continued)
Review of the Derivative:
Derivative: denoted f ( x)
or
y (read f prime of x or y prime)
i) The equation that gives us the slope of the tangent line to the curve at any point x
ii) The equation that give
Calculus Lesson 1.4
Review of Trigonometry: Remember the variety of definitions of the trig functions:
1. Unit circle definitions (a unit circle is a circle with radius 1 unit)
sinA the y-value of where the terminal side of angle A crosses the unit circle
Calculus Lesson 1.5
Exponentials & Logs
An exponential function is a function of the form:
f ( x)
a(b x )
where b>0
if b>1 exponential growth
If b<1 exponential decay
a is the initial value (y-intercept) & b is the growth factor
An exponential is always a
Calculus
Chapter 3
Lesson 1
Let us return to the idea of the slope of the tangent line:
The AVERAGE RATE OF CHANGE is the slope of the secant line between 2 points:
Example: I throw a water balloon out the window and compute the height of the balloon h(t)
AP Calculus Review Worksheet Chapter 4A Name:
Multiple Choice: Select the best answer
No Calculators
1- Let f be a function with a second derivative given by f "(x) = x2 (x3)(x 6). What are
the x-coordinates of the points of inection of the graph of f ?
(
Calculus
Chapter 4 Lesson 4
We know how to identify where a graph is increasing & decreasing but how do we compare
these two graphs?
Both of these functions are always increasing (over their domain) BUT
f(x) increases faster & faster while g(x) increases
Calculus
Chapter 3
Lesson 2
(section 3.1 Continued)
Review of the Derivative:
Derivative: denoted f ( x)
or
y (read f prime of x or y prime)
i) The equation that gives us the slope of the tangent line to the curve at any point x
ii) The equation that give
Calculus Chapter 2
Lesson 3
In Lesson 2 we developed the idea of a limit & saw how to estimate it from a graph or a table.
We better develop some algebraic methods to find a limit.
Here are some rules that always work for limits.
LIMIT RULES:
* lim xc k =
Calculus Chapter 3
Lesson 20
Review for the 2nd test on Chapter 3
No Calculators
Big thing for the test
i) Know How to Take the Derivatives of (Almost) Any Function (or relation)
*Power Functions
*logs & Exponential functions with any base
*Trig Functions
Calculus Chapter 3
Lesson 1
Let us return to the idea of the slope of the tangent line:
The AVERAGE RATE OF CHANGE is the slope of the secant line between 2 points:
Example: I throw a water balloon out the window and compute the height of the balloon h(t)
AP Calculus Test Chapter 3 Part II Derivatives
NOV. 2015 Name:
Multiple Choice: Select the best Answer
No Calculators
7/
MEMO" | a?
3+; 1: ' E
(i l. lime e is y-
hi-D h f 'X
7 r e
a) 1 b) 0 c) e d) e e) undened
#1 Given x) 223$, nd f(0)
:CZ/szf".3rn2. $
Calculus Chapter 2
Lesson 4
In Calculus we discussed a continuous function as one which had no breaks in it. We are ready
for discussing if a function is continuous at a point c.
A function is continuous at c if lim xc f ( x) = f (c)
Examples:
Given the p
Calculus
Chapter 3
Lesson 4
Section 3.2:
Differentiable: We say a function is differentiable at c if it has a derivative at the point c.
f (c h) f (c)
f(x) is differentiable at c, if: limh 0
exists
h
There are several ways that a function may fail to be d
Calculus Chapter 3
Lesson 11
Lesson 10
Applications of the derivative:
A derivative of a function gives us a new function that tells us the instantaneous rate of change at
any given value of x (if the function is differentiable at x )
Identify what the de
Calculus
Chapter 3
Lesson 3
Section 3.2
In section 1 we identified
* that there is a way to find the instantaneous rate of change of a function
* that the instantaneous rate of change of a function was simply a new function:
Unfortunately the process of g
Calculus Chapter 2
Average Rate of Change-
Lesson 1
SLOPE = m
y2
x2
y1
x1
To find the rate of change between any 2 points on a graph, find the slope between the points.
Historical population
Find the average rate of change of the city of Carmel between th
AP Calculus Review Worksheet Chapter 48 Name:
Multiple Choice: Select the best answer
Part A: No Calculators should be used in this section
. k . .
For what value of k Wlll x + have a relative max1mum at x = 2 ?
x
(A) 4 (B) 2 (C) 2 (D) 4 (E) None of these
Calculus Lesson 1.4
Review of Trigonometry: Remember the variety of definitions of the trig functions:
1. Unit circle definitions (a unit circle is a circle with radius 1 unit)
sinA the y-value of where the terminal side of angle A crosses the unit circle
Calculus Chapter 3 Lesson 9
Test on Friday
No Calculators on the TEST
Topics:
*Know what a derivative is, including the definition of a derivative
know both forms of the definition & find the derivative by the definition
*Compute basic derivatives using t