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MATH 110 Module 7 Exam
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MATH 110 Module 7 Exam
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MATH 110 Module 7 Exam |To find horizontal asymptotes - ANSWER take the limit as x approaches -∞ and ∞
(to do this look at dominant powers on top and bottom and remember rules)
Horizontal asymptote rule : If the powers are the same - ANSWER divide the
coefficients
Horizontal asymptote rule : If the power on the bottom is bigger - ANSWER the H.A.
is y=0
Horizontal asymptote rule : If the power on top is bigger - ANSWER the H.A. = DNE
(there is no H.A.)
To find Vertical Asymptotes - ANSWER 1.) Simplify the function to cross out
common factors
2.) After you simplify, set the denominator = 0 and solve
From the First Derivative, f'(x) - ANSWER • Critical Numbers
• Increasing / Decreasing
• Relative Extrema ( Relative Max / Relative Min )
Critical Numbers must satisfy what 2 criteria? - ANSWER • f'(x) = 0 or f'(x) is
undefined
↳In other words, set top and bottom of derivative equal to 0
• Critical Numbers must be in the domain of f(x)
To find Increasing / Decreasing - ANSWER Make a first derivative number line
On first derivative number line : Wherever f'(x) is positive - ANSWER f(x) is
increasing ↗
On first derivative number line : Wherever f'(x) is negative - ANSWER f(x) is
decreasing ↘
To find Relative Extrema - ANSWER Make a first derivative number line
On first derivative number line : Relative Maximum - ANSWER Where f(x) changes
from increasing to decreasing ↗↘
On first derivative number line : Relative Minimum - ANSWER Where f(x) changes
from decreasing to increasing ↘↗
(first derivative) Relative extrema can only occur - ANSWER at critical numbers
From the Second Derivative, f''(x) - ANSWER • Inflection Points
• Concave Up & Concave Down
• The Point of Diminishing Returns
, Inflection Points must satisfy what 3 criteria? - ANSWER • f''(x) = 0 or f''(x) is
undefined
• The concavity must change at that point
• Inflection Points must be in the domain of f(x)
To find Concave Up & Concave Down - ANSWER Make a second derivative
number line
On second derivative number line : Wherever f''(x) is positive - ANSWER f(x) is
Concave Up
On second derivative number line : Wherever f''(x) is negative - ANSWER f(x) is
Concave Down
Point of Diminishing Returns and Inflection Point - ANSWER Mean the same thing
The Point of Diminishing Returns - ANSWER Occurs on an increasing function
where f(x) goes from concave up to concave down
Absolute Extrema - ANSWER Highest or Lowest y-value of the function within a
given interval
For Absolute Extrema: - ANSWER • You will always be given an interval for
absolute extrema questions
• Can occur at the end points of the interval or at the critical numbers
• You do NOT need to set up a first derivative number line (unlike relative extrema)
Steps to Finding Absolute Maximum & Absolute Minimum Values - ANSWER 1.)
Find f'(x)
2.) Find the critical points within the given interval
↳ (ignore critical points outside the interval)
3.) Take the Critical Points from Step 2 and the Endpoints of the interval and set up
an x,y chart by plugging those values back into original function f(x)
4.) Highest y-value is Absolute Maximum and Lowest y-value is Absolute Minimum
For Absolute Maximum & Absolute Minimum, make sure you are giving - ANSWER
y-values, not x-values, when choosing your answer
Optimization - ANSWER In an optimization problem we look for the largest value or
the smallest value that a function can have
Optimization : To maximize or minimize anything - ANSWER Take the Derivative
and set it equal to 0
Optimization : Problems are generally - ANSWER Word problems limited to
business applications or simple geometric applications
Steps to Solve an Optimization Problem - ANSWER 1.) Ask yourself "Overall, what
am I trying to maximize or minimize?"
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