Exam 1

January 29: Limits and how to solve them

Limit Laws

Sum/difference:

Product

Quotient

Power

Roots

Ways to solve limits

Direct Substitution

(if f(x) is a continuous polynomial or rational function) then

lim_x→a f(x) = f(a), so

plug in “a” as the function’s x value

Cancellation

(this is used when utilising direct substitution results in something undefined, like a fraction with 0 in the denominator)

1. factor out the numerator and denominator

2. see if anything can be cancelled out between them

3. cancel it

4. then try direct substitution

Algebraic manipulation

(this can be used when there is a sqrt in the numerator, and when the denominator would be equal to 0 upon attempting direct substitution)

multiply the fraction, whose numerator will be in (A - B) format by (A + B/A + B) to get (A - B)², and then simplify.

NOTE: you will not need to factor out the denominator

One-sided limits

find the lim_x→a from the left side (_x→a-) and from the right side (_x→a⁺), and then see if they are equal

• if they are equal the limit exists.

• if they are not, the limit DNE

Squeeze theorem:

Suppose that f(x) ≤ g(x) ≤ h(x) for all x close enough to a. If lim_x→a f(x) = lim_x→a h(x) = L, then lim_x→a g(x) = L

• this is because g(x) is nestled between f(x) on the left side and h(x) on the right side

February 3: Continuity and continuity theorems

Definition of continuity

a function is continuous at x=a if…

1. f(a) exists; and

2. lim_x→a f(x) = f(a)

Theorems of continuity

Theorem 1:

if f(x), g(x) are continuous at x = a, then so are…

• f(x) ± g(x)

• f(x)*g(x)

• f(g_⁽x_⁾)

• c*f(x)

• f(x)/g(x) as long as g(a) ≠ 0

• nth root of f(x) as long as f(a) > 0 if n is an even number

Theorem 2: polynomial functions

Every polynomial function is continuous everywhere

Theorem 3: rational functions

Every rational function is continuous in its domain

Theorem 4: trig functions

all trig functions are continuous in their domains of definition

Theorem 5: intermediate value theorem

If f(x) is continuous on the interval [a, b], then for every y-value that exists between f(a) and f(b), there exists an x-value

February 5: Infinite limits (VA) and limits at infinity (HA)

Infinite Limits

How to understand of infinite limits ?

• If lim_x→a f(x) = ∞, then when x is arbitrarily close to a, f(x) becomes arbitrarily large

• If lim_x→a f(x) = -∞, then when x is arbitrarily close to a, f(x) becomes arbitrarily small

Vertical Asymptotes

We say that f(x) has a VA at x=a if lim_x→a^± = ± ∞

We can find VA by…

1. identifying values a at which lim_x→a f(x) DNE; or

2. computing one-sided limits for lim_x→a, and if one of these is ±∞, then x=a is a VA

Lim its at Infinity

Definition of limits at infinity

limx→∞ f(x) = L if, when x is arbitrarily large, f(x) becomes arbitrarily close to L

Horizontal Asymptotes

y = L is a HORIZONTAL ASYMPTOTE if lim_x→±∞ f(x) = L

How to identify HA/Limits at infinity

Rule: ¹/_±∞ = 0

NOTE:

• lim_x→0 sin(¹/_x) DNE

• lim_x→∞ ¹/_x sin(x) = 0

For polynomial functions:

Theorem:

• lim_x→∞ xⁿ = ∞

• lim_x→-∞ xⁿ = ∞; if n is even

• lim_x→-∞ xⁿ = -∞; if n is odd

For rational functions:

Given ^{_axn _+…} /^{_bxm _{+ …}}, and that n is the degree of the numerator and m is the degree of the denominator, if…

• n < m, HA @ y = 0

• n = m, HA @ y = ^a/_b

• n > m, there is no HA

February 10: Derivatives and rate of change

Definition of the derivative:

this definition will give you the equation for f’(x)

NOTES

NOTE: if asked to find the equation of the tangent line at a certain point where x = j, plug in j for f’(x)

NOTE: VELOCITY is also an interpretation of derivative

velocity = derivative of position

EXAMPLE with velocity

NOTE: s = speed, t = time, s(t) = position

Given problem: height = 380; equation for distance fallen after t seconds is s(t) = 4.9t² m

1. what is the ball’s velocity after 5 seconds ?

• velocity after 5 seconds can be expressed as v(5), which can be expressed as s’(5), given that we know that velocity is the derivative of position

• find s’(t) or v(t)

• then plug in 5 sec for t

  • this is your answer

2. what is the ball’s velocity when it hits the ground ?

• if the height of the building is 350 m, then the function can be written out like this: s(t) = 380, given that we know that s(t) = position

• find the t

• then find s’(t) which can also be expressed as v(t)

• plug in the time found before into the equation

  • this is your answer

NOTE: point-slope form can also be used to identify the slope if given points, or identify another point if given slope and one ordered pair

• y₁ - y₂ = m (x₁ - x₂)

February 19: The derivative as a function

reminder, definition of derivative:

Differentiability

f(x) is differentiable at x = a if…

• f’(a) exists

  • f’(a) exists if lim h→a exists; so

Continuity

f(x) is continuous at x = a if and only if…

• f(s) exists; and

• lim x→s exists; and

• lim x→s = f(s)

Examples of non-differentiability

if a function is not continuous, it is not differentiable

• Vertical tangent line

• Rapid oscillations

February 24: Quicker ways to compute derivatives

Constant functions

If f(x) = c, then f’(x) = 0

Power functions

f(x) = xⁿ, then f’(x) = n*x^n-1

Constant multiples

(c * f_⁽x_⁾)’ = c * f’(x)

Sum and difference

(f_⁽x_⁾ + g_⁽x_⁾)’ = f’(x) + g’(x)

February 26: Quicker ways to compute derivatives (cont’d)

Product Rule

(f(x) g(x))’ = f’(x) _^x g(x) + f(x) _^x g’(x)

Quotient Rule

(f(x)/g(x))’ = (f’(x) _^x g(x) - f(x) _^x g’(x))/g(x)²

• (lo _x d’hi - hi _x d’lo)/ lo _x lo → lodihi - hidilo/ lolo

Chain Rule

Suppose a(x) = f(g(x)) and f(x), g(x) are both differentiable

then, a’(x) = f’(g(x)) * g(x)

                        |

     compute f’(x) but plug in g’(x) as x

NOTE: f(x) is the OUTER function and g(x) is the INNER function

NOTE: trig function derivatives

• (sin x)’ = cos x

• (cos x)’ = -sin x