- dansmath > lessons >
**calculus 1** **(Part 1 - Limits & Derivatives)**- Limits
- Differential Calculus (Check out the ANIMATION!)
- Integral Calculus (updated 3/99)
- Vector Calculus
- (c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.
- (top of page)

Limits (Sequences, functions, graphs)What do the numbers 1/1, 1/2, 1/3, 1/4, 1/5, . . . get closer and closer to? It may be clear that they approach zero, so we say the

limitis 0. The nth term is 1/n, so the notation is lim (n->oo) 1/n = 0. (The -> means "approaches"; the oo is a cheesy infinity symbol.)The same idea is to consider the function f(x) = 1/x . Then as x -> oo , f(x) -> 0 , so again we say lim (x->oo) f(x) = lim (x->oo) 1/x = 0. On the graph we'd have a horizontal asymptote at y = 0 since the output values approach 0 as the graph goes off to the right. (Click here to review functions or graphing.)

Now what if x -> a and we see if f(x) approaches any specific value L. If it does, then we say that lim (x->a) f(x) = L. By definition x->a means x is near a but not equal to a.

Example 1:Let f(x) = x + 3 . Then as x -> 3, f(x) -> 3 + 3, so lim (x->3) [x+3] = 6.

Example 2:Let g(x) = (x^2 - 9) / (x - 3) . By algebra, we haveg(x) = (x + 3)(x - 3) / (x - 3) and

ifx =/= 3 then we can cancel, sog(x) = x + 3

if x =/= 3.Notice that f(3) = 6 but g(3) is undefined.The f(x) from example 1 has domain "all real numbers," but the g(x) from example 2 has domain "all reals except 3," so they're different functions. But the limit as x -> 3 is the same in both cases: lim (x->3) f(x) = lim (x->3) g(x) = 6. Even though g(3) is undefined (it'd be 0 / 0) g(x) still has a limit (of 6), since x -> 3 implies x =/= 3.

Example 3:Some other interesting limits:lim (x->0) [(sin x) / x] = 1

lim (x->0) [(1 + x)^(1/x)] = e = 2.71828 approx

lim (n->oo) [F(n+1) / F(n)] = (1 + Sqrt(5)) / 2 = 1.61805 approx.

The F(n) is the nth Fibonacci number of the sequence 1, 1, 2, 3, 5, 8, 13, . . .

Differential Calculus

**Tangent Lines & Derivatives . . . . . .**(top of page)- On a straight line graph, y = m x + b, the slope is constant; it's equal to m
- no matter where you are on the graph.
- For the
**slope**we figure (rise)/(run) = m.

But on a parabola, like y = x^2, the direction keeps changing, so we'd expect that the "slope" doesn't stay constant. But how do you figure out the slope of a curve?

**Example 1:**The slope on the parabola y = x^2 is zero at the origin since the curve is- horizontal there. At a different (x, y) location, what's the slope?

Let's take (3,9) as an example; if we join up (3,9) and a nearby point on the curve, say (3.1,9.61), and do "rise/run": (draw yourself a picture)

.

slope= (y2 - y1) / (x2 - x1) = (9.61 - 9) / (3.1 - 3) = 0.61 / 0.1 =6.1If we take a closer point (3.02,3.02^2) = (3.02,9.1204) then the slope is

.

slope= (9.1204 - 9) / (3.02 - 3) = 0.1204 / 0.02 =6.02In fact for any h, the slope between (3,9) and (3+h,(3+h)^2) will be

.

slope= ((3+h)^2 - 9) / (3+h - 3) = (6h + h^2) / h =6 + hThe line joining two points on a curve is called a

secant line. The slope is msec.As h gets smaller and the points get closer together, the slope of the secant line approaches 6. The line it approaches is called the

tangent line. The slope is mtan.

- Here the parabola y = x^2 has a slope of 6 at (3,9) because the limit
- (as h goes to 0) of 6 + h is 6.
**Example 2:**Do the same thing at a general (x,y) point on y = f(x) = x^2:- join up (x,x^2) and (x+h,(x+h)^2) and figure out the slope between them:

. slope = msec = ((x+h)^2 - x^2) / (x+h - x) = (2xh + h^2) / h = 2x + h.

As h goes to 0, this becomes mtan = 2x. The expression for "

the slope at any x" is a new function, derived from f(x), called thederivativeof f, anddenoted f'(x).Here we have:

if f(x) = x^2 then f'(x) = 2x.Checking this for our old point (3,9) we see if x = 3 then f'(x) = 2x = 2(3) = 6. Ok!

Here's a cool Mathematica/ QuickTime/GIFBuilder animation showing the

**secant lines**that approach the**tangent line**. The tangent line is the LIMIT of the secant lines, and the slope**mtan**is the limit of the slopes**msec**.Note that the slope between

**P**and**Q**"settles down" to a value**mtan**, which is the derivative of f at x, where x = x-coord of P.- (top of page)
**Differentiation Rules!**. . . . .- Another notation for f'(x)
is (
**f(x)**)**.** - If
**y = f(x)**then**y' = f'(x) = (y) = dy/dx.** - Notice from above we had
**(x^2) = 2x**. What's the rule for the derivative of a power? - If we had
**f(x) = x^3**then f'(x) = ??? Let's use the limit definition of derivative. **(x^3)**= lim (h->0) [(x+h)^3 - x^3] / h- = lim (h->0) [x^3 + 3 x^2 h + 3 x h^2 + h^3 - x^3] / h
- = lim (h->0) [3 x^2 h + 3 x h^2 + h^3] / h
- = lim (h->0) [3 x^2 + 3 x h + h^2]
- = 3 x^2 + 3 x(0) + 0^2
**= 3 x^2****In general if f(x) = x^n**(works for n = integer, fraction, or irrational!)**then****Power Rule :****(x^n) = n x^(n-1)****Sum Rule : [ f(x) + g(x) ] = f(x) + g(x) = f'(x) + g'(x)****Constant Multiple Rule: [ c f(x) ] = cf(x) = c f'(x).****Some special functions and their derivatives:****(sin x) = cos x , (cos x) = - sin x ,****(e^x) = e^x , (ln x) = 1 / x .****Ex. (x^100) = 100 x^99 ;(x^2 + 5 x) = 2 x + 5 ;****(****x ) = (1/2) x ^ (-1/2) = 1 / ( 2****x ) ;****(5 sin x - 3 cos x) = 5 cos x + 3 sin x.**- We might have a product or quotient of functions:
**Product Rule :****(f(x) g(x))**= f(x)g(x) + g(x)f(x)**= f(x) g'(x) + g(x) f'(x)**

**Quotient Rule :****(f(x) / g(x))**= [ g(x)f(x) - f(x)g(x) ] / (g(x)^2)**= [ g(x)****f'(x) - f(x)****g'(x) ] / (g(x)^2)**

- Here's what you do with a composition of functions:
**Chain Rule :(f(g(x))**= f'(g(x))g(x)**= f'(g(x)) g'(x)****Example:****(sin(3x))**= cos(3x)(3x)**= 3 cos(3x)****(sin^3(x))**= [(sin x)^3]- = 3 [ (sin x)^2 ](sin
x)
**= 3 sin****^2 x cos x** **Applications of Derivatives**. (4/00)

Related RatesYou guessed it -- define some variables, try to relate their rates (of change); we get a relation between their derivatives; technically known as a "differential equation."

Example:Spoze you blow 30 cu.in. of air per sec into a spherical balloon. How fast is the radius increasing when the balloon is 6 inches in diameter?

**Solution:**Ask the important questions:- * What are the variables involved?
- V = volume, r = radius, D = diameter, t = time.
- ** What's the relation among them?
- V = (4/3) Pi r^3; the volume formula for a sphere.
- (Click here for lots of area and volume formulas)
- *** Now take deriv's with respect to time t: (ok, that's not a question)
- d/dt [V] = d/dt [(4/3) Pi r^3] = (4/3) Pi d/dt [r^3] ;
- dV/dt = (4/3) Pi r^3 * 3 r^2 dr/dt = 4 Pi r^2 dr/dt ; this is the chain rule.
- *v Then plug in what you know: D = 6 ==> r = 3, dV/dt = 30 :
- 30 = 4 Pi (3^2) dr/dt ; solve for what you wanna know:
- dr/dt = 30/(36 Pi) ~=~ 0.265 in/sec.

Max/Min ProblemsThe philosophy here is to optimize some quantity Q that can depend on more than one thing (variable). Find a relation (constraint) and use it to get the Q in terms of one variable; then set the first derivative equal to zero and solve!

- (Oh, I was supposed to coax YOU into discovering all that.)

Example:Of all the rectangles of area A, which has the shortest diagonals?

**Approach:**Ok, use x = width, y = height; then xy = A; we want to minimize D = -/(x^2 + y^2) ;- we can sub in y = A/x since A is constant. Also we can minimize D^2 and D will be smallest.
- D^2 = x^2 + (A/x)^2 ; now it's in terms of one variable; take the deriv. w.r.t. x :
- (D^2) = (x^2 + (A/x)^2) = (x^2 + (A^2) x^(-2))
- . . . . . . . . . . = 2x + A^2 (-2 x^(-3)) = 2x - (2 A^2)/(x^3) =set= 0 ;
- 2x = 2 A^2 / x^3 ; x^4 = A^2 ; x = -/A ; y = A/x = -/A ;
**Analysis:**The rectangle's a square, Captain Kirk. And the -/( . ) means square root.

**Go on to Integral Calculus****Well, that's it for now. Check back often for new stuff!****Click below for other topics, or visit the ask dan page!****[ vector calculus | top of page | lessons index]**+ Basic Skills + Arithmetic, Prealgebra, Beginning Algebra + Precalculus + Intermed.Algebra, Functions & Graphs, Trigonometry + Calculus + **You Are Here****1. Limits, Differential Calc**2. Integral Calc, Vector Calc+ Beyond Calculus + Linear Algebra, Diff Equations + Other Stuff + Statistics, Geometry, Number Theory **[ home | info | meet dan | ask dan | dvc ]**This site maintained by B & L Web Design, a division of B & L Math Enterprises.