Sec 1.8 The Derivative as a Rate of Change
ovfea+4
l. Slope: The derivative of x) is a function that gives the slope of the
graph of x) at the point (x, x)
4
tax
2. Rate of Change: The derivative of x
Sec 1.5 Differentiabilitx and Continuity
Not all functions are differentiable at all values in their domain.
Functions will n_ot have derivatives at points where they have sharp
corners discontinui "e
Sec 1.7 More About Derivatives
A. Notation
our
Up to now we have always been using x as an input variable.
We could use any letter we choose. (ex
1
M" x/
4 d I all set
If f(z)= sz Jl4z~s ,then[f(z)]
Sec 1.6 Some Rules for Differentiation
i[Ic-f(x)]= k-gmm
I. Constantmultiple rule, for k a constant:
II. Sum/Difference ruliw d'[f(x) 4'8(x)] " :x[f(x)li' [8 (15)]
HI. General Power rule: r l5 6 5
OSU Mth .241 Midterm 2 1
Oregon State University
Midterm 2 Math 241, FALL 2016
Name OSU ID it
(Print) V
\ E
Instructor: Peter Argyres
Circle your recitation time: 8 9 10 11 12 1 2 3 4
Instructions:
Do
Sec 5.4 Logistic Function
The logistic function was named in 1844/45 by Pierre Verhuist in
relation to population growth.
The. initial stage of growth is approximately exponential, then, as the
popula
Sec 3.2 The Chain Rule and the General Power Rule
Recall the notation we use. for composing functions:
3x1
if f(x)=m and g(X)=x/;,then:
_ 3U? -.\
f(g(x)= 1W?) -
x_r3X\ , 3x\
gm )lvj(><+z) ><+;L
EX
Sec 1.4 Limits and the Derivative
Limit of a function lim f(x) = L
x->c
read: _ the limit, as x approaches 0, of f of x is L
Idea: we want to see what value (number) the function, x), is
getting close
Sec 5.1 Exponential Growth and Decay
Population growth: When modeling population growth situations we
assume that the rate of change of the opulation is proportional to the
WWWWW/Wl/W
size of the popu
Living Beyond
Our Means
NATURAL ASSETS AND
HUMAN WELL-BEING
Statement from the Board
MILLENNIUM ECOSYSTEM ASSESSMENT
Key Messages
Everyone in the world depends on nature and ecosystem services to prov
National Geographic Magazine - NGM.com
Page 1 of 7
Published: April 2010
The Burden of Thirst
If the millions of women who haul water long distances had a faucet by their door, whole
societies could b
Conservation Magazine - When Worlds Collide
Feature
When Worlds Collide
Climate change will shuffle the deck of plants, animals, and ecosystems in ways we've only begun to imagine
By Douglas Fox
Janua
I JC
Sec 4.2 The Ex onential Function: y = e
What is the derivative of the function f (x ) = 2 x ?
Recall what. its graph looks like: I )
x
What is the equation of the line tangent to the graph of
f
Sec 4.5 The Derivative of the Natural L0 Function
of
. mm = em) a x 1
Recall that e x so dx d[ ]_._ dx [ ]
But we could also use the chain rule to perform the differentiation,
in that case we get: I!
Sec 1.3 The Derivative
Imagine the graph of a function, f (x) , where a tangent line to
the graph exists at all points on the graph.
We saw from the example in the last section that it is possible to
Oregon State University
Midterm 2 - Math 241, FALL 2015^
OSU ID #
Name
(Print)
Instructor: Peter Argyres
I f you want this test back you must:
Circle your recitation instructor's name:
Dwight Holland
MTH 241 Derivatives-graphic LAB D Name: KEY
Recitation Time: 1 2 3 4
Below is the graph of a polynomial lnction on the domain of all real numbers.
Estimate the answers to the following questions based