CALCULUS STUDY GUIDE:CHAPTER 3: DIFFERENTIATION

3.1) Tangents and the Derivative at a Point

The slope of the curve y=f(x) at the point P(xo, f(x0)) is the number

o m= limh->0[f(x0+h)-f(x0)]/h *Provided this limit exists*

The tangent line to the curve at P is the line through P with this slope

The normal line to the curve at P is the line through P with the negative reciprocal of this slope

The derivative of a function f at a point xo, denoted f'(x0), is

o f'(x0)= limh->0[f(x0+h)-f(x0)]/h *Provided this limit exists*

Interpretations of the Limit of the Difference Quotient

o limh->0[f(x0+h)-f(x0)]/h Slope of the graph of y=f(x) at x=x0

Slope of the tangent to the curve y=f(x) at x=x0

Rate of change of f(x) with respect to x at x=x0

Derivative f'(x0) at a point

3.2) The Derivative as a Function

The derivative of the function f(x) with respect to the variable x is the function f' whose value at x is

o f'(x)= limh->0[f(x+h)-f(x)]/h Alternate Formula for the Derivative

o f'(x)= limz->x[f(z)-f(x)]/(z-x) Derivative of the Reciprocal Function

o (d/dx)(1/x)=-(1/x2), X=/=0

Theorem #1o If f has a derivative at x=c, then

f is continuous at x=c

3.3) Differentiation Rules

Derivative of a Constant Function If f has the constant value f(x)=c, then

o (df/dx)= (d/dx)(c)= 0

Power Ruleo (d/dx)(xn)= nxn-1

For all x where powers xn and xn-1 are defined

Derivative Constant Multiple Rule If u is a differentiable function of x, and c

is a constant, theno (d/dx)(cu)= c(du/dx)

Derivative Sum Rule If u & v are differentiable functions of x,

then their sum u+v is differentiable at every point where u & v are both differentiable. At such points:

o (d/dx)(u+v)=(du/dx)+(dv/dx)

Derivative Product Rule If u & v are differentiable at x, then so is

their product uv:o (d/dx)(uv)= u(dv/dx)+v(du/dx)

Derivative Quotient Rule If u & v are differentiable at x and if

v(x)=/=0, then the quotient u/v is differentiable at x:o (d/dx)(u/v)=[(v)(du/dx)-u(dv/dx)]/v2

How to Read Symbols for Derivatives y'= y prime y''= y double prime (d2y/dx2)= d squared y dx squared y'''= y triple prime y(n)= y super n (dny/dxn)= d to the n of y dx to the n Dn= D to the n

3.4) The Derivative as a Rate of Change

The instantaneous rate of change with respect to x at x0

o f'(xo)= limh->0[f(x0+h)-f(x0)]/h *Provided the limit exists*

(Instantaneous) velocity is the derivative of position with respect to time. If a body's position at time t is s=f(t), then velocity at time t is

o v(t)=(ds/dt)= lim∆t->0[f(t+∆t)-f(t)]/∆t Speed is the absolute value of velocity

Acceleration is the derivative of velocity with respect to time. If a body's position at time t is s=f(t), then its acceleration at time t is

o a(t)=(dv/dt)=(d2s/dt2)

Jerk is the derivative of acceleration with respect to t

o j(t)=(da/dt)=(d3s/dt3)

3.5) Derivatives of Trig Functions (d/dx)(sinX)= cosX (d/dx)(cosX)= -sinX (d/dx)(tanX)= sec2X (d/dx)(cotX)= -csc2X (d/dx)(secX)= secXtanX (d/dx)(cscX)= -cscXcotX

3.6) The Chain Rule

Theorem #2: The Chain Rule If f(u) is differentiable at the point u=g(x)

and g(x) is differentiable at x, then the composite function (f o g)(x)= f(g(x)) is differentiable at x, and o (f o g)'(x)=f'(g(x))g'(x)

Leibniz's Notationo (dy/dx)=(dy/du)(du/dx)

*Where dy is evaluated at u=g(x)*

3.7) Implicit DifferentiationSteps1. Differentiate both sides of the equation

with respect to x, treating y as a differentiable function of x

2. Collect terms with (dy/dx) on one side of the equation and solve for (dy/dx)

3.8) Related RatesRelated Rates Problem Strategy1. Draw a picture and name the variables

and constants2. Write down the numerical info (In terms

of the symbols you've chosen)3. Write down what you're expected to find

(Usually a rate expressed as a derivative)

4. Write an equation that relates the variables

o You may have to combine two or more equations to relate the variable whose rate you want to the variables whose rates you know

5. Differentiate with respect to to Then express the rate you want

in terms of the rates and variables whose values you know

6. Evaluateo Use known values to find the

unknown rate

3.9) Linearization and DifferentialsIf f is differentiable at x=a, then the approximating function o L(x)= f(a)+f'(a)(x-a)

Is the linearization of f at a

The approximation L(x)~f(x) of f by L is the standard linear approximation of f at a.o The point x=a is the center of the

approximationLet y=f(x) be a differentiable function. The differential dx is an independent

variable The differential dy iso dy=f'(x)dx

Change in y=f(x) near x=a If y=f(x) is differentiable at x=a and x

changes from a to a+∆x, the change ∆y in f is given by

o ∆y=f'(a)∆x + (Epsilon)(∆x) *In which Epsilon->) as x->0*

Sensitivity to ChangeTrue Estimated

Absolute Change

∆f=f(a+dx)-f(a) df=f'(a)dx

Relative Change

(∆f)/[f(a)] (df)/[f(a)]

Percentage Change

((∆f)/[f(a)])(100) ((df)/[f(a)])(100)

4.1) Extreme Values of Functions

Let f be a function with domain D. Then f has an absolute max value on D at point C if o f(x) < f(c) *For all x in D*

and an absolute min value on D at c ifo f(x) > f(c) *For all x in D*

Theorem #1: The Extreme Value Theorem If f is continuous on a close interval

[a,b], then f attains both an absolute max value, M, and absolute min value, m, in [a,b]

o There are numbers, x1 & x2, in [a,b] with f(x1)=m and f(x2)=M, and m<f(x)<M for every other x in [a,b]

A function f has a local max value at a point c w/in its domain D if f(x)<f(c) for all x=D lying in some open interval containing c

A function f has a local min value at a point c w/in its domain D if f(x)>f(c) for all x=D lying in some open interval containing c

Theorem #2: The First Derivative Theorem for Local Extreme Values

If f has a local max or min value at an interior point c of its domain, & f is defined at c, then

o f'(c)=0

An interior point of the domain of a function f where f' is zero or undefined is a critical point of f

How to Find the Absolute Value Extrema of a Continuous f on a Finite Closed Intervalo First evaluate f at all critical points and

endpointso Then take the largest & smallest of

these values