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Calculus I Notes

MATH 52-154/Richards/10.1.07

The Derivative

The derivative of a function f at p is denoted by f(p) and is informally defined by

f(p)= the slope of the f-graph at p.

Of course, the above description only applies when f has a well-defined slope at p. If this is the

case, the magnified f-graph should eventually look linear near the point (p, f(p)).

2 2 4 6 8

50

50

3.5 4.0 4.5 5.0

40

42

44

46

48

50

52

p, fp

3.9 4.0 4.1 4.2

49.0

49.5

50.0

50.5

51.0

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Calculus I/Richards 2

In turn, this visual description brings us to a simple way to approximate f(p): find the slope of

a secant line passing through (p, f(p)) and another nearby point (x, f(x)).

p, fp

x, fx

rise

run

fx fp

x p

3.9 4.0 4.1 4.2

49.0

49.5

50.0

50.5

51.0

Remembering that slope is rise

run, we find that the secant line slope is

msec =f(x) f(p)

xp f(p).

If f has a derivative at p, then we expect the above approximation to improve by choosing x

closer and closer to p (with x = p). This is where the limit enters the picture...

The Real Deal: A Precise Definition of the Derivative. Suppose f is a function defined

on an open interval containing a number p. The derivative of f at p is denoted by f(p) and is

defined by

f(p) = limxp

f(x) f(p)xp

provided this limit exists.

Note: Ifx = p + h, then h = xp and h 0 as x p. This yields the alternative form

f(p) = limh0

f(p + h) f(p)

h

provided this limit exists.

Now it becomes clear: a primary motivation for exploring limits earlier in the semester was to set

the stage for the derivative...

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Calculus Under Construction: A Derivative Knowledge Base and Skill Set

We are developing the skills to solve the following types of problems:

Generate a qualitatively correct graph of f from a given graph of f or f.

Identify local extrema, points of inflection, and intervals on which a function is increasing,

decreasing, concave up, or concave down by investigating its first/second derivative.

Use the limit definition of the derivative to directly find derivatives of the following types of

functions: f(x) = xn (n a positive integer or n = 1/2, 1/3), f(x) = (ax + b)/(cx + d), sin(x) &

cos(x).

Use the limit definition of the derivative to verify the product and quotient rules.

Know the derivatives of polynomial, rational, power, trig., exponential, logarithmic, inverse

trig. functions.

Use the established derivative rules (power, sum, product, quotient, chain) to find the derivatives

of functions built from polynomial, rational, power, trig., exponential, logarithmic, inverse trig.

functions.

Find an equation of a tangent line (or secant line) to the graph of a function.

Interpret the meaning of a derivative in a specific applied problem. (e.g., velocity, marginal

cost, ...).

Use tangent lines to approximate more complicated functions.

Use derivatives to find extreme values of a given function.

Use implicit differentiation to find unknown rates of change by relating them to known rates of

change.

More to come...

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Limits: Approaching an Ultimate Destination

Intuitive Definitions of Limit. Suppose f is a function defined on an open interval con-

taining a number p except possibly at p. Then f is said to have a limit L R at p iff f(x) can

be made to be arbitrarily close to L by choosing x sufficiently close to p with x = p. In this case

we writelimxp

f(x) = L.

This can also be translated as f(x) approaches L as x approaches p with x = p.

Intuitive Definitions of One-Sided Limits.

limxp+

f(x) = L can be translated as f(x) approaches L as x approaches p from above.

limxp

f(x) = L can be translated as f(x) approaches L as x approaches p from below.

Connection: Suppose f is a function defined on an open interval containing a number p except

possibly at p. It follows that limxp

f(x) = L iff limxp+

f(x) = L = limxp

f(x).

Limit Laws. Suppose f and g are functions defined on an open interval containing a number

p except possibly at p If f and g both have real limits at p, then

limxp

(f(x) + g(x)) = limxp

f(x) + limp

g(x)

limxp

(f(x) g(x)) = limxp

f(x) limxp

g(x)

limxp

(f(x) g(x)) = limxp

f(x) limxp

g(x)

limxp

(f(x)/g(x)) = limxp

f(x)/ limxp

g(x),

where the last equation requires that limxp

g(x) = 0.

Squeeze Theorem. Suppose g(x) f(x) h(x) for all x near p (except perhaps at x = p).

If g and h have the same limit L at p, then f is squeezed into having the limit L at p. That is,

if limxp

g(x) = L = limxp

h(x), then limxp

f(x) = L.

(If the Limit Laws arent enough, it may be time to put on the Squeeze. )

Precise Definition of Limit. Suppose f is a function defined on an open interval containing a number p except

possibly at p. Then f is said to have a limit L R at p iff for every > 0 there is some > 0 such that L < f(x) < L+ for

all x = p satisfying p < x < p + . (Okay, maybe this definition seems a little technical, but it is something every calculus

student should ponder if only to refine ones understanding and appreciate the simplicity of the above intuitive definition.) If

you are still reading this, then you probably anticipated the fact that one-sided limits also have similar precise definitions.

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Continuous Functions

As weve seen in several examples, a functions limit at a point p and its value at p dont always

agree. When they do, we celebrate the property by giving it a special name:

Suppose f is a function defined on an open interval containing a number p. Then f is said to be

continuous at p iff limxp f(x) = f(p).Connection to Limits: By the Limit Laws, we find that if f and g are continuous at p then so are

f + g, f g and f /g (if g(p) = 0). This immediately expanded our library of known elementary

continuous functions. Recall that we often applied this reasoning as a strategy for calculating

limits by first simplifying a given limit to one involving a function in our library of continuous

functions.

Continuity on Intervals: If f is continuous at each element of an open interval (a, b), then

f is said to be continuous on (a, b). Iff is continuous on (a, b) and satisfies limxa+ f(x) = f(a) and

limxb

f(x) = f(b), then f is said to be continuous on the closed interval [a, b].

This terminology came along at just the right time. Now we can state (and understand) theorems

like the Intermediate Value Theorem and the Extreme Value Theorem.

Intermediate Value Theorem. Suppose f is continuous on [a, b] and is any number

strictly between f(a) and f(b) (i.e., f(a) < < f(b) or f(a) > > f(b)). Then there is some

input c (a, b) such that f(c) = .(The IVT serves as guarantee that solutions to certain equations exist.)

Extreme Value Theorem. Suppose f is continuous on [a, b]. Then f has an absolute

maximum value and an absolute minimum value on [a, b].

(Now the interesting question: How do we find the absolute extrema for a given function that is

continuous on [a, b]?)

Useful Miscellanea

an bn = (a b)(an1 + an2b + ... + abn2 + bn1) (for positive integers n 2)

limh0

sin(h)

h= 1

limh0

cos(h) 1

h= 0

sin(A + B) = sin(A)cos(B) + cos(A) sin(B)

cos(A + B) = cos(A)cos(B) sin(A) sin(B)

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