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Chapter 2: Limits and Derivatives
2.1 The Tangent and Velocity Problems
2.2 The Limit of a Function
Limits: lim f(x) = L
x→a
One-sided Limits: lim f(x) = L
x→a−
x=aisavertical asymptote of y = f(x) if at least one one-sided limit as x approaches a is ±∞.
2.3 Calculating Limits Using the Limit Laws
If lim f(x) and lim g(x) exist (particularly not ±∞):
x→a x→a
lim[f(x)+g(x)] = lim f(x)+ lim g(x)
x→a x→a x→a
lim[f(x)−g(x)] = lim f(x)− lim g(x)
x→a x→a x→a
lim[cf(x)] = c lim f(x) where c is a constant
x→a x→a
lim[f(x)g(x)] = lim f(x)· lim g(x)
x→a x→a x→a
f(x) lim f(x)
lim[ ] = x→a if limx→ag(x) 6= 0
x→a g(x) lim g(x)
h x→a i
n n
lim[f(x)] = lim f(x) where n is a positive integer
x→a x→a
p q
lim n f(x) = n lim f(x), n is a pos. int. (if n is even, assume lim f(x) > 0)
x→a x→a x→a
Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a and lim f(x) =
lim h(x) = L, then lim g(x) = L. x→a
x→a x→a
2.5 Continuity
Afunction f(x) is continuous at a if lim f(x) = f(a).
x→a
Polynomials, exponentials, logarithms, roots, trig functions, inverse trig functions and rational func-
tions are all continuous at each point in their domains.
If f, g are continuous at a, c constant, then f +g, f −g, fg, cf, fg, f if g(a) 6= 0 are continuous at a.
g
If g is continuous at a and f is continuous at g(a), then f ◦ g is continuous at a.
Intermediate Value Theorem: Suppose that f is continuous on [a,b] and let N be any number between
f(a) and f(b), where f(a) 6= f(b). Then there exists a number c in (a,b) such that f(c) = N.
2.6 Limits at Infinity: Horizontal Asymptotes
y = L is a horizontal asymptote of the curve y = f(x) if lim = L or lim =L.
x→∞ x→−∞
2.7 Derivatives and Rates of Change
The tangent line to y = f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is equal to f′(a), the
derivative of f at a.
Point-slope formula: The equation of a line with slope f′(a) at the point (a,f(a)) is y − f(a) =
f′(a)(x−a)
The following terms mean the same thing: the derivative, the slope of the line tangent to the curve,
and the instantaneous rate of change.
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2.8 The Derivative as a Function
Definition of the derivative: f′(x) = lim f(x+h)−f(x) (provided the limit exists)
h→0 h
Differentiation is the process of taking a derivative, and a function f is differentiable at a if f′(a)
exists.
If f is differentiable at a then f is continuous at a.
Given y = f(x), we can denote its derivative as f′(x), y′, d f(x), or d y = dy (Leibniz notation).
dx dx dx
⊲ Chapter 2 Notes:
• Possible Questions:
Evaluate the limit of a given function.
From a graph of a function, determine the limit of the function.
Find horizontal or vertical asymptotes of a function using limits.
Find constants that make a particular piecewise function continuous.
Show there exists roots or certain values of a function using the intermediate value theorem.
Find the limit of a function using the Squeeze Theorem.
Find the equation of the tangent line to a function at a given point.
From a graph of a function, sketch the graph of its derivative.
Find the derivative of a function using the definition of the derivative.
• Notation notes, and things to be careful of:
Make sure you write limits correctly. Do not drop the limit until you get rid of the limiting variable.
Also, make sure to write = in each line.
Be careful when you say f(x) = a function where you’ve canceled out terms, the functions may not be
equal.
If a limit is ±∞, it technically does not exist. Infinity is not a number.
±
Always make sure to figure out if a function goes to +∞ or −∞ when taking the limit using 0 .
Do not say something equals something that is either undefined or indeterminate. This includes
indeterminate forms and numbers divided by 0. (We fudge this rule a bit when we say a limit = ±∞.)
Although there are several equivalent definitions of the derivative, your life will be easier if you use the
h→0definition in practice.
DONOT:
lim f(x) = 0 lim f(x) = ∞ lim = f(x) lim f(x) = nonzero #
x→a x→a x→a x→a ±
0 ∞ 0
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Chapter 3: Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
d (c) = 0 (where c is any constant)
dx
d (xn) = nxn−1 (where n is any constant)
dx
d x x d x x
dx(e ) = e dx(a ) = a lna
If f and g are both differentiable and c is constant, then
d [cf(x)] = cf′(x) d [f(x)±g(x)] = f′(x)±g′(x)
dx dx
3.2 The Product and Quotient Rules
If f and g are both differentiable, then
d [f(x)g(x)] = f(x)g′(x)+g(x)f′(x) d f(x) = g(x)f′(x)−f(x)g′(x)
dx dx g(x) (g(x))2
3.3 Derivatives of Trigonometric Functions
d d d 2
dx(sinx) = cosx dx(cosx) = −sinx dx(tanx) = sec x
d (cscx) = −csccotx d (secx) = sectanx d (cotx) = −csc2x
dx dx dx
3.4 The Chain Rule
If g is differentiable at x and f is differentiable at g(x), then the composite F = f ◦ g defined by
F(x) = f(g(x)) is differentiable at x, and F′ is given by F′(x) = f′(g(x)) · g′(x).
Also written as d f(g(x)) = f′(g(x))·g′(x) or dy = dy · du
dx dx du dx
3.5 Implicit Differentiation
Implicit Differentiation: Differentiate both sides of the equation y = f(x) with respect to x and then
solve the resulting equation for y′ = dy.
dx
d (arcsinx) = √ 1 d (arccosx) = −√ 1 d (arctanx) = 1
dx 1−x2 dx 1−x2 dx 1+x2
d (arccscx) = − √ 1 d (arcsecx) = √ 1 d (arccotx) = − 1
dx 2 dx 2 dx 1+x2
x x −1 x x −1
3.6 Derivatives of Logarithmic Functions
d (lnx) = 1 d (log x) = 1 d (ln|x|) = 1
dx x dx b xlnb dx x
Logarithmic Differentiation: Take natural logarithms of both sides of an equation y = f(x) and use
the logarithm laws to simplify, and then differentiate implicitly with respect to x to solve for y′.
3.7 Rates of Change in the Natural and Social Sciences
′
If s(t) is the position function of a particle at time t, then v = s (t) represents the instantaneous
velocity, and a = v′(t) = s′′(t) represents the acceleration.
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3.8 Exponential Growth and Decay
dy kx
The differential equation dx = ky only has solutions of the form Ce , where C is a constant. More
kx
precisely, the solutions are y(x) = y(0)e .
Herewesaythat“thegrowth/decayrateisproportionaltothesize/mass”or“therelativegrowth/decay
rate is constant.”
The half-life is the time required for half of a quantity to decay.
⊲ Chapter 3 Notes:
• Possible Questions:
Find the derivative of a function using the derivative rules.
Find the equation of the line tangent to a curve at a given point.
Find the equations of lines that are parallel/perpendicular to the tangent line at a particular point.
Find the x-values at which the tangent line of f(x) is parallel/perpendicular to a given line.
Use implicit/logarithmic differentiation to find a derivative.
Given a position function of a physical object (projectiles, particles, etc), answer questions about its
position, velocity, and acceleration.
What is its velocity/acceleration after t seconds?
When does it hit the ground?
What is its velocity/acceleration when it is at a certain position?
How far did it travel?
When does it reach maximum height, and what is the max height?
When is it at rest?
If an experimental quantity has constant growth/decay rate (perhaps half-life),
Find a formula for the amount at time t.
Find the relative growth/decay rate given some data points.
Find the amount after a certain amount of time has passed.
Find the rate of growth/decay after a certain amount of time has passed.
Find the time at which the amount is a certain number.
• Notation notes, and things to be careful of:
Use derivative notation correctly, especially Leibniz notation!
Know how to simplify using log and exponent rules, and be on the lookout for ”disguised constants.”
Know how to recognize when to use which derivative rule.
Product, quotient, chain rule - recognize a product, a quotient, or a composite function
Implicit differentiation - if the function is not written as y = f(x) = something involving only x
Logarithmic differentiation - often will have powers involving x or complicated products or quotients.
Remember your logarithm and exponent rules!
d n n−1 d x x
Note that the derivatives dx(x ) = nx and dx(a ) = a lna are different.
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