Calculus Chapter 1 Flashcards

Questions and Answers

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?

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?

Set y equal to 0.

How do you find y-intercepts?

Set x equal to 0. Signup and view all the answers

How do you write an equation for a line that passes through (2,0) with a slope of 1?

Use the y = mx + b equation, plug in x and y along with the slope, then solve for b, and reinsert into the equation. Signup and view all the answers

What does ƒ(time) mean?

It means the function of time. Signup and view all the answers

What does ƒ(x) represent?

It means the function of x. Signup and view all the answers

When asked to express the sum of two numbers as a function of the smaller number, what should you do?

Solve for the number we are trying to get the function of and plug it into the original equation. Signup and view all the answers

What is a limit?

The y value of a function or equation as you approach a certain x value. Signup and view all the answers

For a function to have a limit, what must be true?

The limit as we approach the x value from the left and the right side must be the same number. Signup and view all the answers

How do you find the limit?

Plug in the x value we are approaching into the equation. Signup and view all the answers

If the limit is 0/0, what do we have?

Indeterminate form. Signup and view all the answers

How do we continue to solve a limit problem with an indeterminate form?

Try to factor the equation and cancel out some factors. Signup and view all the answers

How do we continue to solve a limit problem where we have an undefined value, like 2/0 or 1/0?

Draw the graph or try to factor. Signup and view all the answers

Find the limit of (x² + 2) as x approaches negative infinity.

The limit is positive infinity. Signup and view all the answers

Find the limit of (1/x) as x approaches infinity.

The limit approaches 0. Signup and view all the answers

What trick helps when finding infinite limits?

Divide everything by the higher power in the denominator. Signup and view all the answers

When confused on whether an infinity limit is positive or negative, what should you do?

Just plug in a random huge number for x. Signup and view all the answers

The only time when the domain of the function isn't all real numbers is when?

We have fractions and radicals in the equation. Signup and view all the answers

How do you solve for the domain when we have fractions?

Set the denominator equal to zero and solve. Signup and view all the answers

How do you solve for the domain when we have radicals?

Set what is inside the radical to ≥ 0. Signup and view all the answers

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.