Chapter 2 Calculus: Hughes-Hallett

The Derivative

Continuity of y = f(x)

A function is said to be continuous if there are no “breaks” in its graph.

A function is continuous at a point x = a if the value of f(x) L, a number, as x a for values of x either greater or less than a.

Continuous Functions-

The function f is continuous at x = c if f is defined at x = c and

The function is continuous on an interval [a,b] if it is continuous at everypoint in the interval.

If f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval. (A theorem.)

).()(lim cfxfcx

Definition of Limit-

Suppose a function f, is defined on an interval around c, except perhaps not at the point x = c.

The limit of f(x) as x approaches c: is the number L (if it exists) such that f(x) is as close to L as we please whenever x is suffici-ently close to c (but x c).

In Symbols:

L)x(flimcx

.|L)x(f|then|cx|0if

,0,0,,

,0,0,L)x(flimcx

Properties of Limits-

Assuming all the limits on the right hand side exist:

cxbb

xgprovidedxg

xf

xg

xf4.

xgxfxgxf3.

xgxfxgxf2.

tantconsabxfbxbf

cxcx

cxcx

cx

cx

cxcxcx

cxcxcx

cxcx

lim.6lim.5

0)(lim,)(lim

)(lim

)(

)(lim

))(lim))((lim()()(lim

)(lim)(lim)()(lim

,)(lim)(lim.1

Limits at Infinity-

If f(x) gets as close to a number L as we please when x gets sufficiently large, then we write:

Similarly, if f(x) approaches L as x gets more and more negative, then we write:

Lxfx

)(lim

Lxfx

)(lim

Average Rate of Change-

The average rate of change is the slope of the secant line to two points on the graph of the function.

h

)x(f)hx(f

xx

)x(f)x(f

xx

yym 11

12

12

12

12sec

The Derivative is --

Physically- an instantaneous rate of change.

Geometrically- the slope of the tangent line to the graph of the curve of the function at a point.

Algebraically- the limit of the difference quotient as h 0 (if that exists!). In symbols:

x

)x(f)xf(x

h

)x(f)hf(x)x('f

dx

df

dx

dy 11

0x

11

0h1 limlim

First Derivative Interpretation-

If f’ > 0 on an interval, then f is increasing over the interval.

If f’ < 0 on an interval, then f is decreas-ing over the interval.

Derivative Symbols:

If y = f(x) = then each of the following symbols have the same meaning:

And at a particular point, say x = 2, these symbols are used:

3x3x3

3x3yDfD'y)x('fdx

)3x3x(d

dx

)x(df

dx

df

dx

dy 2xx

3

93)2(3)2(fD)2('y)2('fdx

dy 2x

2x

Basic Formulas (1):

Derivative of a constant: If f(x) = k, the f’(x) = 0, k - a constant

Derivative of a linear function: If f(x) = mx + b, then f’(x) = m

Derivative of x to a power: 1nn nx)x('fthen,x)x(fIf

Second Derivative Interpretation-

If f’’ > 0 on an interval, then f’ is increasing, so the graph of f is concave up there.

If f’’ < 0 on an interval, then f’ is decreasing, so the graph of f is concave down there.

If y = s(t) is the position of an object at time t, then:

Velocity: v(t) = dy/dt = s’(t) = Acceleration: a(t) =

)t(s

vs)t('v)t(''sdt/yd 22

Continuous Functions-

The function f is continuous at x = c if f is defined at x = c and

The function is continuous on an interval [a,b] if it is continuous at everypoint in the interval.

If f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval. (A theorem.)

).()(lim cfxfcx

Continuity of y = f(x)

A function is said to be continuous if there are no “breaks” in its graph.

A function is continuous at a point x = a if the value of f(x) L, a number, as x a for values of x either greater or less than a.

Theorem on Continuity-

Suppose that f and g are continuous on an interval and that b is a constant. Then, on that same interval:

1. bf(x) is continuous.2. f(x) + g(x) is continuous.3. f(x)g(x) is continuous.4. f(x)/g(x) is continuous, provided

` on the interval.0)x(g

Differentiability-

A function f is said to be differentiable at x = a if f’(a) exists.

Theorem: If f(x) is differentiable at a point x = a, then f(x) is continuous at x = a.

Linear Tangent Line Approximation-

Suppose f is differentiable at x = a. Then, for values of x near a, the tangent line approximation to f(x) is:

The expression is called the local linearization of f near x = a. We are thinking of a as fixed, so that f(a) and f’(a) are constant. The error E(x), is defined by and

).)((')()( axafafxf ))((')( axafaf

).)((')()()( axafafxfxE .0

)(lim

ax

xEax