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Write each equation in standard form. Then state whether the graph of
the equation is a parábola, circle, ellipse and hyperbola.
Without writing the equation in standard form, state whether the graph
of each equation is a parábola, circle, ellipse, and hyperbola.
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Cooling towers for nuclear power
plants are often built in the shape
of a ………………
Source:https://math.libretexts.org/TextMaps/Calculus/Book%3A_Calculus_(OpenStax)/12%
3A_Vectors_in_Space/12.6%3A_Quadric_Surfaces
Energy reflects off of the
parabolic reflector and is
collected at the focal point.
(credit: modification of CGP
Grey, Wikimedia Commons)
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Finding Limits Using a Graph
What is a limit?
Calculus involves a major shift in perspective and one of the first shifts
happens as you start learning limits. When I talk about the limit of a
function f(x)f(x) as xx approaches some value, I am not saying “what
is f(x)f(x) at this value” like I might in algebra! Instead, I am interested in what
is happening to f(x)f(x) when xx is close to this value.
lim
𝑥→𝑎
𝑓 𝑥 = 𝐿
“The limit of f(x) as x approaches a is L”
For example:
lim
𝑥→1
(2𝑥2 − 𝑥 − 1)
(𝑥 − 1)
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SPECIAL LIMITS
The number e is defined as a limit. Here is one definition:
lim
𝑥→0
1 + 𝑥
1
𝑋 = 𝑒
A good way to evaluate this limit is make a table of numbers
The number e is the natural base in calculus. Many expressions in calculus
are simpler in base e than in other bases like base 2 or base 10
e = 2.71828182845904509080 · ·
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Big ideas of calculus
Integrals, derivatives and the fact that they’re opposites.
DERIVATIVES USING THE LIMIT DEFINITION
.
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Source: Limits | Chapter 7, The esence of calculus
lim
𝑥→1
𝑆𝑖𝑛(𝜋𝑥)
𝑥2 − 1
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Source: Limits | Chapter 7, The esence of calculus
PROOF IN LIMITS
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L’Hopital’s rule
Example:
lim
𝑥→0
𝑆𝑖𝑛(𝑥)
𝑥
Source: Limits | Chapter 7, The esence of calculus
PROOF IN LIMITS
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INTEGRALS
An integral is a mathematical object that can be interpreted as an area
or a generalization of area. Integrals, together with derivatives, are the
fundamental objects of calculus. Other words for integral include
antiderivative and primitive. The Riemann integral is the simplest
integral definition and the only one usually encountered in physics and
elementary calculus.
A definite integral of a function can be represented as the signed area of the
region bounded by its graph.
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IMPORTANT
Taking the derivative of f(x) may precisely give you g(x), but taking
the antiderivative of g(x) does not necessarily give you f(x) in its
original form
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INTEGRALS OF TRIGONOMETRIC FUNCTIONS
That is, every time we have a differentiation formula, we get an integration formula
for nothing. Here is a list of some of them.
Notice that, quite by chance, we have come up with formulas for the
antiderivatives of sin x and cos x
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Monthly sales of Ocean King Boogie Boards are
given bys(t) = 1,500sin(π/(t − 7)/6) + 2,000,where t is time in
months, and t = 0 represents January 1. Estimate total sales over
the four-month period beginning March 1.
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RIEMANN SUMS AND THE DEFINITE INTEGRAL
RIEMMAN SUMS WITH “INFINITE” RECTANGLES
