Download Study Guide for Chapter 2

Math-M119: Brief Survey of Calculus
Don Byrd, Instructor (revised version)
Study Guide for Text Chapter 2: Rate of Change: The Derivative
Note: items in square brackets “[…]” are relatively unimportant.
2.1 Instantaneous Rate of Change
Instantaneous Velocity
- Use average velocity over an interval of time to estimate instantaneous
- Ex: throw grapefruit straight up
Generally, the shorter the interval, the better the estimate
Defining Instantaneous Velocity Using the Idea of a Limit
Instantaneous Rate of Change
- Instantaneous = in an infinitesimal (infinitely small) time interval
Change in velocity is acceleration
Derivative at a Point
- Def’n: derivative of f at a, written f ’(a), is instantaneous rate of change of f at a
- …it’s the slope of tangent to f at a
Visualizing the Derivative: Slope of the Graph, of the Tangent Line
- Average rate of change as ∆x decreases => secant line approaches tangent line
- Review: value increasing/decreasing vs. slope increasing/decreasing
Does the derivative always exist? = Is there always a tangent at every point in domain?
(And is the domain all real numbers?)
- No! Exx: 1/x ; abs(x) . (For each, WHERE DOES IT NOT EXIST?)
2.2 The Derivative Function
The derivative of a function is another function.
Finding the Derivative of a Function Given Graphically
What Does It Tell Us Graphically?
f ’ > 0 on an interval => f is increasing over the interval
f ’ < 0 on an interval => f is decreasing over the interval
f ’ = 0 on an interval => f is constant over the interval
Finding the Derivative of a Function Given Numerically
Finding the Derivative of a Function Given by a Formula
2.3 Interpretations of the Derivative
Alternative notation:
€
dy
(Liebniz)
dx
1
- NOT a fraction, but kind of like one…
(“infinitesimal” change in y) / (“infinitesimal” change in x)
[ this is interpreting it as a differential ]
Using units to interpret the derivative [ unit analysis, a.k.a. dimensional analysis ]
dy
Notation for value of derivative at a point:
x= nnn
dx
Using the derivative to estimate function values
- From the point where value is desired, can go to right, to left, or both ways
Local Linear Approximation: for x€“near” a
f (x) ≈ f (a) + f ' (a)(x − a)
NB: x – a, not a – x. (WHY?)
€
€
Relative rate of change of y = f(t) at t = a
dy / dt
f ' (a)
=
y
f (a)
[ Marginal cost (p. 115) ]
- Linear & non-linear cost functions
Non-linearity caused by economy of scale, glutted markets, etc.
Marginal cost = slope of cost curve
Marginal revenue = slope of revenue curve
- Average rate of change is approx. = to instantaneous rate of change (derivative)!
2.4 The Second Derivative
Def’n: 2nd derivative is the derivative of the (1st) derivative
- derivative exists only when function and 1st derivative are both differentiable
Notation: for y = f(x), 1st derivative = f ′ =
dy 2
dy
; 2nd derivative = f ′′ = 2
dx
d x
What Does the 2nd Derivative Tell Us?
f ’’ > 0 on an interval => f ’ is increasing over the interval
f ’’ < 0 on an interval€=> f ’ is decreasing over the
€ interval
What does this behavior of 1st derivative mean for the original function?
f ’ increasing => f is bending upward (concave up) (WHY?)
f ’ increasing => f is bending upward (concave up) (WHY?)
- caveat: “going up” means “becoming more positive”; “going down” means
“becoming more negative”!
- common real-life example: f = distance, f ’ = velocity, f ’’ = acceleration
Interpretation as Rate of Change
- actually, rate of change of a rate of change!
- Ex: if you accelerate hard, velocity goes up quickly
- …and distance changes more and more quickly
2
- Specific ex: the 2011 Nobel Prize in Physics went to three astronomers for showing
that, if t = time & s(t) = distance from Earth to far-off supernovas, s’’ is positive =
the expansion of the universe is accelerating
Focus on Theory
[ Def’n: limit ]
[ Notation: the limit as x approaches c of f(x) is L is written: lim f (x) = L ]
x→c
[ Def’n of derivative in terms of limits ]
Def’n: a function f(x) is differentiable at any point x at which the derivative function exists
€
The Textbookʼs built-in Chapter Summary (p. 120)
Rate of change
•
Average, instantaneous
Estimating derivatives
•
Estimate derivatives from a graph, table of values, or formula
Interpretation of derivatives
•
Rate of change, slope, using units, instantaneous velocity
Relative rate of change
•
Calculation and interpretation
Marginality
•
Marginal cost and marginal revenue
Second derivative
•
Concavity
Derivatives and graphs
•
Understand relationship between sign of f’ and whether f is increasing or decreasing.
Sketch graph of f’ from graph of f. Marginal analysis
3
Functions and Derivatives and their Graphs
Function and Its Sign
Graph of f(x)
Sign of function value f(x)
above x axis
+
on x axis
0
below x axis
–
Function vs. Sign of First Derivative
Graph of f(x)
Rate of change of f(x)
Sign of 1st derivative, f ’
going up
positive
+
constant
0
0
negative
–
going down
Function vs. Sign of Second Derivative
Graph of f(x)
Rate of change of f(x)
Sign of 2nd derivative, f ’’
concave up
becoming more positive
+
straight line
0
0
becoming more negative
–
concave down
DAB, rev. 8 Oct. 2011
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