Asymptotes, Continuity, and Theorems

Question 1

Horizontal Asymptotes

Answer 1

If the limit as x approaches positive or negative infinity of a function equals L, then the line y = L is a horizontal asymptote of the function.

Question 2

Vertical Asymptotes

Answer 2

The line x = a is a vertical asymptote of the function if either the limit approaching a on the positive or negative side equals positive or negative infinity.

Question 3

Slant (Oblique) Asymptotes

Answer 3

If the limit of a function over x as x approaches positive or negative infinity equals L, then the line y = Lx is a slant (oblique) asymptote of the function.

Question 4

A function is continuous as point x = a if

Answer 4

1, f(a) exists. 2, the limit of a function as x approaches a exists. 3, the limit of a function as x approaches a equals f(a). Make sure to check both sides of the limit.

Question 5

Instantaneous rate of change

Answer 5

The limit as h approaches 0 of average speed, or the derivative at a given point.

Question 6

IVT

Answer 6

If (f) is a continuous function on a closed interval [a, b] and if ynaught is any value between f(a) and f(b) then ynaught = f(c) for some c in [a, b].

Question 7

Squeeze Theorem

Answer 7

Suppose that g(x) </= f(x) </= h(x) for all x in some open interval containing c, except possibly at x = c itself. Suppose also that the limit of g(x) as x approaches c = the limit of h(x) as x approaches c = L. Then the limit of f(x) as x approaches c equals L.

Question 8

MVT

Answer 8

Under certain conditions you can find a point (c) such that its tangent line is parallel to its secant line if f is continuous over [a,b], f is differentiable over (a,b) and there exists at least one real number in (a,b) such that f’(c) = (f(b)-f(a))/(b-a)

Question 9

Consequences of MVT

Answer 9

1. f’(x) = 0 for a<x<b 2. f(x) = C (constant) for a<x<b 3. If f’(x) > 0 (positive) over (a,b) then f(x) is increasing in (a,b). If f’(x) < 0 (negative) over (a,b) then f(x) is decreasing in (a,b).

Question 10

Rolle’s Theorem

Answer 10

If f is continuous over [a,b], differentiable over (a,b) and f(a) = f(b), then there is at least one number in (a,b) such that f’(c) = 0.

Question 11

Linearization

Answer 11

L(x) = f’(a)(x-a) +f(a), change in L = f’(a)*change in x

Question 12

Newton’s Method

Answer 12

Guess first approximation to a solution of the equation f(x) = 0, and then use the formula to find the 1st, 2nd, 3rd approximation and so on. x(n+1) = x(n) - (f(x(n))/f’(x(n))), if f’(x(n)) does not equal 0.

Question 13

Tangent Line

Answer 13

y - y(0) = m(x - x(0)), where m is the slope (average rate of change, secant line, derivative). Touches the curve at one point.

Question 14

Secant Line

Answer 14

The average rate of change, or slope. Touches the curve at two points, (x(1), y(1)) and (x(2), y(2)). (y(2) - y(1)/x(2) - x(1)) or [f(x2) - f(x1)]/x2 - x1 = travelled distance/duration = change in y/change in x = rise/run

Question 15

Normal Line

Answer 15

If the tangent line is y - y(0) = m(x - x(0)), then the normal line is y - y(0) = (-1/m)(x - x(0)), where m is the derivative/slope.