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Questions and Answers
What is the value of $ ext{lim}_{x o -1} f(x)$ for the function defined as $f(x) = \begin{cases} x^3 & \text{if } x < 1 \ x^2 & \text{if } x \geq 1 \end{cases}$?
What is the value of $ ext{lim}_{x o -1} f(x)$ for the function defined as $f(x) = \begin{cases} x^3 & \text{if } x < 1 \ x^2 & \text{if } x \geq 1 \end{cases}$?
- 0
- -1 (correct)
- 1
- undefined
The limit $ ext{lim}_{t o 2} \frac{t^2 - 4t + 4}{t - 2}$ is equal to 0.
The limit $ ext{lim}_{t o 2} \frac{t^2 - 4t + 4}{t - 2}$ is equal to 0.
False (B)
What is the derivative of $y(x) = 7x^2$ using the definition of a derivative?
What is the derivative of $y(x) = 7x^2$ using the definition of a derivative?
14x
If the slope of the function $y(x)$ is 10, then we can say that the function is _____ at that point.
If the slope of the function $y(x)$ is 10, then we can say that the function is _____ at that point.
Which of the following describes the function $y(x) = \begin{cases} \cos(x) & \text{if } x \geq 0 \ -2x + 1 & \text{if } x < 0 \end{cases}$?
Which of the following describes the function $y(x) = \begin{cases} \cos(x) & \text{if } x \geq 0 \ -2x + 1 & \text{if } x < 0 \end{cases}$?
Find the limit $ ext{lim}_{t o 0} \frac{t^2 - 4t + 4}{t - 2}$.
Find the limit $ ext{lim}_{t o 0} \frac{t^2 - 4t + 4}{t - 2}$.
Match the following functions with their derivatives:
Match the following functions with their derivatives:
The limit as $x$ approaches 1 for the function $f(x)$ is _____ for the second part of the piecewise function.
The limit as $x$ approaches 1 for the function $f(x)$ is _____ for the second part of the piecewise function.
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Study Notes
Assignment Overview
- Due date: 11:59 PM, Wednesday, 8 May 2024
- Submission format: Online via designated link, single PDF file
- File naming convention: Must end with yourUserName.pdf
Limit Calculations
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Task involves finding limits for given functions
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Example function for limits:
- ( f(x) = \begin{cases} x^3 & \text{if } x < 1 \ x^2 & \text{if } x \geq 1 \end{cases} )
- Compute:
- ( \lim_{x \to -1} f(x) )
- ( \lim_{x \to 1} f(x) )
-
Additional limits to evaluate:
- ( \lim_{t \to 2} \frac{t^2 - 4t + 4}{t - 2} )
- ( \lim_{t \to 0} \frac{t^2 - 4t + 4}{t - 2} )
Derivative Calculations
- Derivatives must be calculated using the definition of a derivative, not relying on common derivative tables
- Functions to find derivatives for:
- ( y(x) = 7x^2 )
- ( y(x) = 2x - 10 )
- ( y(x) = x )
Function Slope Inquiry
- The slope of the function ( y(x) ) is given as 10
- Explanation required discussing implications for the behavior of the function
Derivative Existence Analysis
- Analyze a piecewise function:
- ( y(x) = \begin{cases} \cos(x) & \text{if } x \geq 0 \ -2x + 1 & \text{if } x < 0 \end{cases} )
- Objective:
- Determine if there are points where the derivative does not exist
- Justification should be based on logical reasoning rather than graph analysis
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