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Questions and Answers
What is the difference quotient?
What is the difference quotient?
ƒ(x+h) - ƒ(x) / h
What is the process for finding the difference quotient of ƒ(x) = 4 - 5x?
What is the process for finding the difference quotient of ƒ(x) = 4 - 5x?
- Plug in x+h for every x in equation: 4 - 5(x+h) → 4 - 5x - 5h. 2) Plug the new equation for ƒ(x+h) in the difference quotient equation. 3) Plug in ƒ(x) = 4 - 5x for ƒ(x) in the difference quotient equation.
How do you find x-intercepts?
How do you find x-intercepts?
Set y equal to 0.
How do you find y-intercepts?
How do you find y-intercepts?
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How do you write an equation for a line that passes through (2,0) with a slope of 1?
How do you write an equation for a line that passes through (2,0) with a slope of 1?
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What does ƒ(time) mean?
What does ƒ(time) mean?
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What does ƒ(x) represent?
What does ƒ(x) represent?
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When asked to express the sum of two numbers as a function of the smaller number, what should you do?
When asked to express the sum of two numbers as a function of the smaller number, what should you do?
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What is a limit?
What is a limit?
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For a function to have a limit, what must be true?
For a function to have a limit, what must be true?
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How do you find the limit?
How do you find the limit?
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If the limit is 0/0, what do we have?
If the limit is 0/0, what do we have?
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How do we continue to solve a limit problem with an indeterminate form?
How do we continue to solve a limit problem with an indeterminate form?
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How do we continue to solve a limit problem where we have an undefined value, like 2/0 or 1/0?
How do we continue to solve a limit problem where we have an undefined value, like 2/0 or 1/0?
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Find the limit of (x² + 2) as x approaches negative infinity.
Find the limit of (x² + 2) as x approaches negative infinity.
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Find the limit of (1/x) as x approaches infinity.
Find the limit of (1/x) as x approaches infinity.
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What trick helps when finding infinite limits?
What trick helps when finding infinite limits?
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When confused on whether an infinity limit is positive or negative, what should you do?
When confused on whether an infinity limit is positive or negative, what should you do?
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The only time when the domain of the function isn't all real numbers is when?
The only time when the domain of the function isn't all real numbers is when?
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How do you solve for the domain when we have fractions?
How do you solve for the domain when we have fractions?
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How do you solve for the domain when we have radicals?
How do you solve for the domain when we have radicals?
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Study Notes
Difference Quotient
- Defined as ƒ(x+h) - ƒ(x) / h
- A fundamental concept in calculus, used for finding derivatives.
Difference Quotient Example
- For ƒ(x) = 4 - 5x, substitute x+h for x to get 4 - 5(x+h) → 4 - 5x - 5h
- Replace ƒ(x+h) in the difference quotient with the new equation and ƒ(x) appropriately.
Finding x-intercepts
- Set y equal to 0, as x-intercepts occur at points where y = 0.
Finding y-intercepts
- Set x equal to 0, indicating where the y-intercepts exist.
Equation of a Line
- To write a line equation through (2,0) with slope 1, use y = mx + b.
- Plug in known values to solve for b and reformulate the line equation.
Function Notation
- ƒ(time) refers to the function of time.
- ƒ(x) denotes the function of x.
Expressing Sum of Two Numbers
- Solve for the number that represents the value to be eliminated and substitute in the original equation to express as a function.
Limit Definition
- Refers to the y-value of a function as x approaches a specific value.
Conditions for a Limit
- For a function to have a limit, left-hand limit and right-hand limit must coincide at a specific x value.
Calculating Limits
- Substitute the approaching x value directly into the equation to find the limit.
Indeterminate Form
- Occurs when evaluating a limit yields 0/0.
Resolving Indeterminate Form
- Factor the equation to cancel out terms and simplify to obtain a real number.
Handling Undefined Values
- For limits that yield undefined forms (e.g., 2/0 or 1/0), consider graphing or factoring as methods to assess the limit.
Limit at Negative Infinity
- For lim (x² + 2) as x approaches -∞: results in positive infinity since -∞² = +∞ and adding 2 yields positive infinity.
Limit as x Approaches Infinity
- For lim (1/x) as x approaches ∞: the limit approaches 0 as the denominator increases indefinitely.
Infinite Limits Trick
- Divide all terms by the highest power in the denominator for simplifying infinite limits.
Determining Infinity Limits
- To discern if an infinite limit is positive or negative, substitute a large value for x into the function.
Domain Restrictions
- A function's domain isn't all real numbers when fractions or radicals are present.
Domain of Functions with Fractions
- Set the denominator equal to zero and solve, excluding those values from the domain.
Domain of Functions with Radicals
- Set the expression inside the radical ≥ 0, solving for x provides bounds for valid real numbers in the domain.
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Description
Test your knowledge with these flashcards covering key concepts from Calculus Chapter 1. Learn about the difference quotient, how to compute it for a specific function, and the process for finding x-intercepts. Perfect for reviewing foundational calculus concepts.