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https://docs.google.com/document/d/1hL0tbZFOxaQFwAoVeJBQZUTmmnrSCt4FlAUT4RaOQTQ/edit?usp=sharing

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5 Shortcut Rules for Finding Derivatives:

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☆ Limit definition to find the derivative of a function:
- the derivative of a function exists when the value that is being approached from the left equals the value from the right.
- to find the derivative of the function direct substitute, then if you get indeterminate form (0/0) manipulate by factoring, conjugate pairs, or limit properties.

☆ Choose two of the problems below and find the derivative by the limit process. Show your work.

f(x)=x+4 f(x)=(1x2) f(x)=2-x2

☆ Alternate limit definition to find the derivative of the tangent line to a curve at a point:
- If f is defined on an open interval containing a, and m (slope) = lim (as n approaches 0) then (f(x+h) - f(x)) ÷ (x) exists. therefor the line passing through (a, f(a)) with the slope is the tangent line that belongs to the graph at (a, f(a)).

☆ Choose one of the problems below and find the derivative at the given point. Show your work.

f(t)=t2+3, (-2, 7) f(x)=x-1, (5, 2) f(x)=1x+1, (0, 1)

☆ What does g'(8)=10 mean?
-  g'(8)=10 means that the limit of the function g is 10, as x goes to 8. This limit value is found using the alternate form.

☆  What does it mean for a function to be differentiable?
- For a function to be differentiable the limit exists, which means the limit from the left has to = the limit from the right on an open interval.

☆  How do you algebraically show a function is differentiable?
1) find the derivative from the left and the right.
2) make sure there are no sharp corners.
3) make sure there are no vertical tangent lines
*differentiable means continuity, BUT continuity does NOT mean differentiable.

P. 272 (#97-98) Show your work.

f

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Polynomial, Rational, and Piecewise Functons

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Polynomial, Rational, and Piecewise Functons

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The approach taken in the problem was to place 2.99 and 3.01 in for the x value in the original limit equation to see what the product would be for both the left and the right of the limit. The product stays consistent with the original limit for 3, therefore the limit exists, and there will be no discontinuity in the graph.

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Removable and Nonremovable  Discontinuity: 

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Removable and Nonremovable  Discontinuity: 

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☆A function with a non-removable discontinuity at x=4.                                  ☆A function with a removable discontinuity at x=-4. 

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☆A function with a non-removable discontinuity at x=4.                                  ☆A function with a removable discontinuity at x=-4. 

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☆ Rational functions do not always have verticle asymptotes. There could be a hole instead of a verticle asymptote of a factor of the numerator cancels a factor of the denominator.

P. 253 (#34-36). Show your work.

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☆ What is Continuity?
A function f(x) is continuous at x + a if....
1) f(a) is defined
2) limit f(x) exists
x → a
3) f(a) = lim f(x)
x → a
☆  Non-Removeable versus Removable Discontinuity:

Removeable - hole

Non-Removable - jump, infinite

Choose 3 problems from the textbook - one polynomial function, one rational function and one piecewise function. Then, discuss the continuity of each function and where it may not be continuous. Show your work.

☆ Intermediate Value Theorem:

if function f(x) is continuous on [a,b] - closed interval - than f(x) hits every y-value between f(a) and f(b). f(c) exists between a and b.

Conditions:

1) closed interval, continuous.

2) identify f(a) and f(b).

3) check desired value is between f(a) and f(b).

☆IVT Example Problem↓

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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☆ 2 problems
1) Solved with a table:
2) Solved graphically:

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☆ Limit Properties:
1) lim      c = c
x → a

the limit from the left equals the limit from the right.

2) lim c (f(x))
x → a

evaluate the limit, then multiply by c.

3) lim f(x) x lim g(x)
x → a
= lim f(x) x lim g(x)
x → a      x → a

apply   x → a to all the limits being multiplied.

4) lim f(x) +/- g(x)
x → a
= lim f(x) +/- g(x)
x → a          x → a

apply   x → a to all the limits being added or subtracted.

5) lim f(x)/g(x)
x → a
= lim f(x)    /   lim g(x)
x → a.          x → a

apply   x → a to all the limits being divided (denominator cannot = 0)

6) lim g(f(x)
x → c
=   g(lim f(x))
x → c

Break it down: evaluate f(x), then evaluate g( the limit you

x → c,   found with f(x).
x → c

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3.2 Finding Limits Graphically and Numerically

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3.2 Finding Limits Graphically and Numerically

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☆ The notation of
lim f(x)=25
x→8
means what is the limit at 25 (y-axis) as x approaches 8 (on the x-axis).

☆If f(2)=4, can you conclude anything about the limit of f(x) as x approaches 2? The limit could exist because the left is equal to the right. There also could be a hole if there is nothing coming from the left and right.
☆If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? f(2) means the limit exists, but it might not be defined.

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☆ What is calculus?

Calculus is finding a point at a specific instant of time, in the second, versus an average or over a period.