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Final

MATH 1200 Final: Limits, Derivatives, and Graphing
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4 Pages
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Fall 2016

Department
Mathematics
Course Code
MATH 1200
Professor
Matthew Demers
Study Guide
Final

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MATH 1200 8 Dec 2016
LIMITS
Tangent Lines
A tangent line is a line that locally touches a function at only one point. To find the slope of a tangent line, at a
given point a:
m= lim
h0
f(a+h)f(a)
h
Anormal line is the line perpendicular to the tangent; they have a slope of 1/m.
Limit Rules
Assuming that limxaf(x)and limxag(x)both exist, where a, c R.
1. limxa[f(x)±g(x)] = limxaf(x)±limxag(x)
2. limxacf(x) = c·limxaf(x)
3. limxaf(x)g(x) = limxaf(x)·limxag(x)
4. limxa
f(x)
g(x)=limxaf(x)
limxag(x)
5. limxah(f(x)) = h(limxaf(x)), for h(x)continuous.
Factor Theorem
If f(p) = 0 for any polynomial f(x),(xp)is a factor of f(x).
Ex. Find limx2
2x4+ 17x3+ 31x262x168
6x5+ 5x4153x3+ 8x2+ 660x400.
If you directly substitute x= 2 into the above function, you get 0
0. This means that (x2) is a factor for
both polynomials in the function. Factor (x2) out of each polynomial, using synthetic division, then try direct
substitution again.
One-sided Limits
Functions can sometimes be discontinuous at certain points, this is especially true for piece-wise functions. A limit
exists only when a function is continuous, this is determined by evaluating a limit approaching xfrom both the left
and right.
limxaf(x) = Lmeans the limit of f(x)approaching xfrom the left.
limxa+f(x) = L+means the limit of f(x)approaching xfrom the right.
If L̸=L+, the limit doesn’t exist.
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Description
MATH 1200 8 Dec 2016 L IMITS Tangent Lines A tangent line is a line that locally touches a function at only one point. To find the slope of a tangent line, at a given point a: f▯a ▯ h▯ ▯ f▯a▯ m ▯ ▯h▯▯ h A normal line is the line perpendicular to the tangent; they have a slope of ▯▯=m. Limit Rules Assuming that ▯▯▯ f▯x▯ and ▯▯▯ g▯x▯ both exist, where a;c ▯ R. x▯a x▯a 1. ▯▯x▯a ▯f▯x▯ ▯ g▯x▯▯ ▯ ▯▯x▯a f▯x▯ ▯ ▯▯▯x▯a g▯x▯ 2. ▯▯x▯a cf▯x▯ ▯ c ▯ ▯▯x▯a f▯x▯ 3. ▯▯x▯a f▯x▯g▯x▯ ▯ ▯▯▯ x▯a f▯x▯ ▯ ▯x▯a g▯x▯ f▯x▯ ▯▯▯x▯a f▯x▯ 4. ▯▯x▯a ▯ g▯x▯ ▯▯▯x▯a g▯x▯ 5. ▯▯x▯a h▯f▯x▯▯ ▯ h▯▯▯▯ x▯a f▯x▯▯, for h▯x▯ continuous. Factor Theorem If f▯p▯ ▯ ▯ for any polynomial f▯x▯, ▯x ▯ p▯ is a factor of f▯x▯. ▯ ▯ ▯ ▯x ▯ ▯▯x ▯ ▯▯x ▯ ▯▯x ▯ ▯▯▯ Ex. Find ▯▯x▯▯ ▯x ▯ ▯x ▯ ▯▯▯x ▯ ▯x ▯ ▯▯▯x ▯ ▯▯▯ . ▯ If you directly substitute x ▯ ▯ into the above function, you . This means that ▯x ▯ ▯▯ is a factor for ▯ both polynomials in the function. Factor ▯x ▯ ▯▯ out of each polynomial, using synthetic division, then try direct substitution again. One-sided Limits Functions can sometimes be discontinuous at certain points, this is especially true for piece-wise functions. A limit exists only when a function is continuous, this is determined by evaluating a limit approaching x from both the left and right. ▯▯▯ ▯f▯x▯ ▯ L ▯eans the limit of f▯x▯ approaching x from the left. x▯a ▯▯▯x▯a +f▯x▯ ▯ L ▯eans the limit of f▯x▯ approaching x from the right. If L ̸▯ L , the limit doesn’t exist. ▯ ▯ 1 MATH 1200 8 Dec 2016 Limits to Infinity When evaluating limits to infinity, it’s important to keep the following in mind: ▯ ▯▯; ▯a▯ < ▯ ▯ ▯▯▯ a ▯ ▯; a ▯ ▯ ▯▯▯ ▯ ▯ ▯; a > ▯ n▯▯ ▯∄; a ▯ ▯▯ n▯▯ na ▯ ∄; ▯a▯ > ▯ D ERIVATIVES First Principles The derivative is related closely to the tangent of a function. The derivative of a function at point a yields a number by ′ f▯a ▯ h▯ ▯ f▯a▯ f ▯a▯ ▯ ▯▯▯ h▯▯ h provided the limit exists. The derivative of an entire function is ′ f▯x ▯ h▯ ▯ f▯x▯ f ▯x▯ ▯ h▯▯ h Notations ′ dy f ▯x▯ ▯ Newton’s f prime of x ▯ Liebniz notation f ▯ Newton’s dot notation dx Differentiation Rules These rules make differentiation easier, so you don’t need to use first principles. 1. The constant function rule: 5. The product rule: f▯x▯ ▯ k; k ▯ R f▯x▯ ▯ uv f ▯x▯ ▯ ▯ f ▯x▯ ▯ ▯uv▯ ▯ u v ▯ uv′ 2. The power rule: 6. The quotient rule: f▯x▯ ▯ x ; n ▯ R f▯x▯ ▯ u ′ n▯▯ v f ▯x▯ ▯ nx ( u ′ u v ▯ uv′ f ▯x▯ ▯ ▯ ▯ 3. The constant multiple rule: v v f▯x▯ ▯ k ▯ g▯x▯; k ▯ R; g▯x▯ is differentiable 7. The power of a function rule
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