Chapter 3: Applications of Derivatives

§ 7:

Linear approximation and differentials

Tangent line to y = f(x) at (x₀,f(x₀) : L(x) = f(x₀)+f^¢(x₀)(x-x₀)

f(x) » L(x) for x near x₀

Ex.: [Ö27] » 5+[1/2·5](27-25), using f(x) = Öx

(1+x)^k » 1+kx, using x₀=0

Df = f(x₀+Dx)-f(x₀), then f(x₀+Dx) » L(x₀+Dx) translates to

Df » f^¢(x₀) ·Dx

differential notation: df = f^¢(x₀) dx

So Df » df, when dx = dx is small

In fact, Df-df = (diffrnce quot -f^¢(x₀))Dx = (small)·(small) = really small, goes like (Dx)²

§ 8:

Newton's method

Idea: tangent line to the graph of a function ``points towards'' a root of the function

Roots of (tangent) lines are easy to find!

L(x) = f(x₀)+f^¢(x₀)(x-x₀) ; root is x₁ = x₀-[(f(x₀))/(f^¢(x₀))]

Now use x₁ as starting point for new tangent line; keep repeating!

x_n+1 = x_n-[(f(x_n))/(f^¢(x_n))]

Basic fact: if x_n approximates a root to k decimal places, then x_n+1 tends to approximate it to 2k decimal places!

BUT:

Newton's method might find the ``wrong'' root: Int Value Thm might find one, but N.M. finds a different one!

Newton's method might crash: if f^¢(x_n) = 0, then we can't find x_n+1 (horizontal lines don't have roots!)

Newton's method might wander off to infinity, if f has a horizontal asymptote; an initial guess too far out the line will generate numbers even farther out.

Newton's method can't find what doesn't exist! If f has no roots, Newton's method will try to ``find'' the function's closest approach to the x-axis; but everytime it gets close, a nearly horizontal tangent line sends it zooming off again!

Chapter 4: Integration

§ 1:

Antiderivatives

F is an antiderivative (or (indefinite) integral) of f if F^¢(x) =f(x).

Notation: F(x) = òf(x) dx ; it means F^¢(x)=f(x)

``the integral of f of x dee x''

Basic list:

òxⁿ dx = [(xⁿ⁺¹)/(n+1)] + C (provided n ¹ -1)

òsin(kx) dx = [(-cos(kx))/k] + C

òcos(kx) dx = [sin(kx)/k] + C

òsec² x dx = tanx + C

òcsc² x dx = -cotx + C

òsecxtanx dx = secx + C

òcscxcotx dx = -cscx + C

Most differentiation rules can be turned into integration rules (although some are harder than others; some will even wait until Calc II !)

Basic integration rules: sum and constant multiple rules are easy to reverse

k=constant

òk·f(x) dx = kòf(x) dx

ò(f(x)±g(x) dx = òf(x) dx ± òg(x) dx

§ 3:

Integration by substiution

if g(x) = u, then [d/dx]f(g(x))=[d/dx]f(u) = f^¢(u) [du/dx]

so òf^¢(u) [du/dx] dx = òf^¢(u) du = f(u)+c

òf(g(x)) g^¢(x) dx ; set u = g(x)

then du = g^¢(x) dx , so òf(g(x)) g^¢(x) dx = int f(u) du , where u = g(x)

Example: òx(x+2-3)⁴ dx ; set u = x²-3, so du=2x dx . Then

òx(x+2-3)⁴ dx = [1/2]ò(x+2-3)⁴2x dx =[1/2]òu⁴ du |_{u = x²-3} =

[1/2][(u⁵)/5]+c |_{u = x²-3} = [((x²-3)⁵)/10]+c

The three most important points:

1. Make sure that you calculate (and then set aside) your du before doing step 2!

2. Make sure everything gets changed from x's to u's

3. Don't push x's through the integral sign! They're not constants!

§ 4:

Estimating things with sums

Sigma notation: å_{i = 1}ⁿ a_i = a₁+¼a_n ; just add the numbers up

formal properties:

å_{i = 1}ⁿ ka_i = kå_{i = 1}ⁿ a_i

å_{i = 1}ⁿ (a_i±b_i) = å_{i = 1}ⁿ a_i ±å_{i = 1}ⁿ b_i

Some things worth adding up:

length of a curve: approximate curve by a collection of straight line segments

length of curve » å(length of line segments)

distance travelled = (average velocity)(time of travel)

over short periods of time, avg. vel. » instantaneous vel.

so distance travelled » å(inst. vel.)(short time intervals)

E.g., s(t)=position, v(t)=velocity, use velocity 4 times per second

dist. travelled = s(10)-s(5) » å_{i = 1}²⁰ v(5+[i/4]) ([1/4])

average value of a function

average of n numbers: add the numbers, divide by n

for a function, add up lots of values of f, divide by number of values

avg. value of f » [1/n]å_{i = 1}ⁿ f(c_i)

§ 5:

Definite integrals

Idea: approximate region b/w curve and x-axis by things whose areas we can easily calculate:

rectangles!

We define the area to be the limit of these sums as the number of rectangles goes to ¥ (i.e., the width of the rectangles goes to 0), and call this the definite integral of f from a to b:

ò_a^b f(x) dx  =  lim_n®¥ å_{i = 1}ⁿ f(c_i)Dx_i

When do such limits exist?

Theorem If f is continuous on the interval [a,b], then ò_a^b f(x) dx exists.

(i.e., the area under the graph is approximated by rectangles.)

§ 6:

Properties of definite integrals

Basic properties of definite integrals:

ò_a^a f(x) dx =0

ò_b^a f(x) dx = -ò_a^b f(x) dx

ò_a^b kf(x) dx = kò_a^b f(x) dx

ò_a^b f(x)±g(x) dx =ò_a^b f(x) dx ± ò_a^b g(x) dx

ò_a^b f(x) dx + ò_b^c f(x) dx = ò_a^c f(x) dx

If m £ f(x) £ M for all x in [a,b], then

m(b-a) £ ò_a^b f(x) dx £ M(b-a)

More generally, if f(x) £ g(x) for all x in [a,b], then

ò_a^b f(x) dx £ ò_a^b g(x) dx

Average value of f : formalize our old idea!

avg(f) = [1/(b-a)]ò_a^b f(x) dx

Mean Value Theorem for integrals: If f is continuous in [a,b], then

there is a c in [a,b] so that f(c) = [1/(b-a)]ò_a^b f(x) dx

§ 7:

The fundamental theorem of calculus

ò_a^x f(t) dt = F(x) is a function of x.

F(x) = the area under the graph of f, from a to x.

Fund. Thm. of Calc (# 1): If f is continuous, then F^¢(x) = f(x)

(F is an antiderivative of f !)

Since any two antiderivatives differ by a constant, and F(b) = ò_a^b f(t) dt, we get

Fund. Thm. of Calc (# 2): If f is continuous, and F is an antiderivative of f, then

ò_a^b f(x) dx = F(b)-F(a) = F(x) |_a^b

Ex: ò₀^p sinx dx = (-cosp)-(-cos0) =2

Building antiderivatives:

F(x)=ò_a^x Ö{sint} dt is an antiderivative of f(x) = Ö{sinx}

G(x) = òx2x3 [Ö(1+t2)] dt = F(x3)-F(x2), where

F^¢(x) = [Ö(1+x²)], so G^¢(x) = F^¢(x³)(3x²)-F^¢(x²)(2x)...

§ 8:

substitution and definite integrals

ò_a^b f(x) dx really means ò_{x = a}^{x = b} f(x) dx

and we remember to change all of the x's to u's!

Ex: ò₁² x(1+x²)⁶ dx; set u = 1+x², du = 2x dx . when x = 1, u = 2; when x = 2, u = 5 ; so

ò₁² x(1+x²)⁶ dx = [1/2]ò₂⁵ u⁶ du = ...