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Math 2053
Calculus I
Highlights
(Chapter 3)
Definition of Extrema
• Let f be defined on an interval I containing c.
1. f(c) is the minimum of f on I if f(c) ≤ f(x) for all
x in I.
2. f(c) is the maximum of f on I if f(c) ≥ f(x) for
all x in I.
• The minimum and maximum of a function on an interval are
the extreme values, or extrema (the singular form is
extremum), of the function on the interval. The minimum and
maximum of a function on an interval are also called the
absolute (or global) minimum or maximum on the interval.
Definition of Extrema
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Definition of Relative Extrema
• 1. If there is an open interval containing c on which
f(c) is a maximum, then f(c) is called a relative
maximum of f, or you can say that f has a
relative maximum at (c, f(c)).
• 2. If there is an open interval containing c on which
f(c) is a minimum, then f(c) is called a relative
minimum of f, or you can say that f has a relative
minimum at (c, f(c)).
Note: The plural of relative maximum (minimum) is
relative maxima (minima).
Critical Numbers
• Let f be defined at c. If f ’(c) = 0 or if f is not
differentiable at c, then c is a critical number of f.
• If f has a relative minimum or relative maximum at
x = c, then c is a critical number of f.
• In other words, relative extrema of a function occur
only at the critical numbers of the function.
Critical Numbers
a “cusp”
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Guidelines for Finding
Extrema on a Closed Interval
• To find the extrema of a continuous function f on a
closed interval [a, b], use the following steps:
1. Find the critical number(s) of f in (a, b).
2. Evaluate f at each critical number in (a, b).
3. Evaluate f at each endpoint of [a, b]. I.e., Find
f(a) and f(b).
4. The least of these values is the minimum. The
greatest is the maximum.
Rolle’s Theorem
• Let f be continuous on the closed interval [a,b]
and differentiable on the open interval (a,b).
If f(a) = f(b), then there is at least one number
c in (a, b) such that f ’(c) = 0. Therefore, if f is
continuous on [a,b], differentiable on (a,b), and
f(a) = f(b), then there must be at least one
x-value between a & b at which the graph of f
has a horizontal tangent.
Rolle’s Theorem
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The Mean Value Theorem
• Rolle’s Theorem can be used to prove the Mean
Value Theorem — and vice versa. Some consider
these to be the most important theorems in calculus,
along with the highly related Fundamental Theorem
of Calculus.
• If f is continuous on the closed interval [a, b] and
differentiable on the open interval (a, b), then there
exists a number c in (a, b) such that
f(b) – f(a)
f ' c =
b–a
Increasing and
Decreasing Functions
• A function f is increasing on an interval if for any two
numbers in the interval, implies that
As x values increase, y values increase, and the graph
moves up.
• A function f is decreasing on an interval if for any two
numbers in the interval, implies that
As x values increase, y values decrease, and the graph
moves down.
• A function is said to be strictly monotonic on an interval if it
is either increasing or decreasing on the entire interval.
Derivative Test for Increasing
and Decreasing Functions
• Let f be a function that is continuous on the closed
interval [a, b] and differentiable on the open interval
(a, b).
1. If f’(x) > 0 for all x in (a, b), then f is increasing
on [a, b].
2. If f’(x) < 0 for all x in (a, b), then f is decreasing
on [a, b].
3. If f’(x) = 0 for all x in (a, b), then f is constant
on [a, b].
Derivative Test for Increasing
and Decreasing Functions
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Guidelines for Increasing and
Decreasing Functions
• Let f be continuous on the interval (a, b). To find the
open intervals on which f is increasing or decreasing,
use the following steps:
1. Locate the critical numbers of f in (a, b), and use
these numbers to determine test intervals.
2. Determine the sign of f’(x) at one test value in
each of the intervals.
Guidelines for Increasing and
Decreasing Functions
3. By the sign of f’, determine whether f is
increasing or decreasing on each interval.
• These guidelines are also valid if the interval (a, b) is
replaced by an interval of the form –, b , a, , or –, .
The First Derivative Test
• Let c be a critical number of a function f that is continuous on
an open interval I containing c. If f is differentiable on the
interval, except possibly at c, then
1. If f’(x) changes from negative to positive at c, then f has
a relative minimum at (c, f(c)).
2. If f’(x) changes from positive to negative at c, then f has
a relative maximum at (c, f(c)).
3. If f’(x) is positive on both sides of c or negative on both
sides of c, then f(c) is neither a relative minimum nor a
relative maximum.
The First Derivative Test
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Definition/Test for Concavity
• Let f be differentiable on an open interval I.
The graph of f is concave upward on I if f’ is increasing on
the interval and concave downward on I if f’ is decreasing
on the interval.
• Let f be a function whose second derivative exists on
an open interval I.
1. If f”(x) > 0 for all x in I, then the graph of f is concave
upward in I.
2. If f”(x) < 0 for all x in I, then the graph of f is concave
downward in I.
Concavity
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Concave up is like a “cup”; concave down is like a “crown”.
Points of Inflection
• Let f be a function that is continuous on an open
interval and let c be a point in the interval. If the
graph of f has a tangent line at this point (c, f(c)), then
this point is a point of inflection of the graph of f if
the concavity of f changes from upward to downward
(or downward to upward) at the point.
• If (c, f(c)) is a point of inflection of the graph of f,
then either f ”(c) = 0 or f ” does not exist at x = c.
Points of Inflection
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Points of inflection can occur where f”(0) = 0 but (0,0) is not a point of
f”(x) = 0 or f” does not exist. inflection
The Second Derivative Test
• Let f be a function such that f ’(c) = 0 and the second
derivative of f exists on an open interval containing c.
1. If f ”(c) > 0, then f has a relative min at (c, f(c)).
2. If f ”(c) < 0, then f has a relative max at (c, f(c)).
3. If f ”(c) = 0, then the test fails. That is, f may
have a relative maximum, a relative minimum,
or neither. In this case, you can use the First
Derivative Test.
The Second Derivative Test
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Definition of a
Horizontal Asymptote
The line given by y = L
is a horizontal asymptote
of the graph of f if
lim f(x) = L or lim f(x) = L
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The graph of a function
of x can have at most two
horizontal asymptotes.
Guidelines for Finding Limits
at ± for Rational Functions
• 1. If the degree of the numerator is less than the degree of the
denominator, then the limit of the rational function is 0 (and
the horizontal asymptote (H.A.) is y = 0).
• 2. If the degree of the numerator is equal to the degree of the
denominator, then the limit of the rational function is the
ratio of the leading coefficients (and the H.A. is y = ratio of
leading coefficients).
• 3. If the degree of the numerator is greater than the degree of
the denominator, then the limit of the rational function does
not exist (and there is no H.A.).
Guidelines for Analyzing the
Graph of a Function
• 1. Determine the domain and range of the
function.
• 2. Determine the intercepts, asymptotes, and
symmetry of the graph.
• 3. Locate the x-values for which f ’(x) and
f ”(x) either are zero or do not exist. Use the
results to determine relative extrema and
points of inflection.
Optimization Problems
1. Identify all given quantities and quantities to be
determined. If possible, make a sketch.
2. Write a primary equation for the quantity that is
to be maximized or minimized. (Refer to
geometry formulas inside the front cover of our
text.)
3. Reduce the primary equation to one having . . .
Optimization Problems
3. . . . a single independent variable. This may
involve the use of secondary equations relating
the independent variable of the primary equation.
4. Determine the feasible domain of the primary
equation. That is, determine the values for which
the stated problem makes sense.
5. Determine the desired maximum or minimum
value by using our calculus techniques.
Volume of a Box Problem
Find the “optimal”
open box with a
square base and a
fixed surface area
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the dimensions of
the box with
maximum volume.
Volume of a Box Problem
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Newton’s Method
Let f(c) = 0, where f is differentiable on an open
interval containing c. Then, to approximate c,
use the following steps:
Make an initial estimate x1 that is close to c. (A
graph is helpful.)
Determine a new approximation using
f x n
xn + 1 = xn –
f ' x n
Newton’s Method (cont’d)
f x n
xn + 1 = xn –
f ' x n
If the value is within the desired accuracy, let xn+1
serve as the final approximation. Otherwise, return
to step 2, and calculate a new approximation.
Each successive application of this procedure is
called an iteration.
Newton’s Method
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The x-intercept of the tangent line approximates the zero of f.
Tangent Line Approximation
Consider a function f that is
differentiable at c. The equation
for the tangent line at the point
(c, f(c)) is given by
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y = f(c) + f ’(c)(x – c)
and is called the tangent line
approximation (or linear
approximation) of f at c.
Differentials
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Differentials
• Let y = f(x) represent a function that is
differentiable on an open interval containing x.
The differential of x (denoted by dx) is any
nonzero real number. The differential of y
(denoted by dy) is dy = f’(x)dx.
x = x – c
y dy = f '(x) dx
Error Propagation Formulas
The main idea is using dy to approximate ∆y.
∆x represents the error in measurement (of height, radius,
etc.); ∆y also represent the error in measurement (of
area, circumference, volume, etc.)
y dy
- is called the relative error of the measurement
y y
Relative error is often converted to percent error.
Differential Formulas
• Let u and v be differentiable functions of x.
Constant Multiple: d[cu] = c du
Sum or Difference: d[u ± v] = du ± dv
Product: d[uv] = u dv + v du
u v du – u dv
Quotient: d =
v v2
