What are the three properties that define continuity of a function f(x) at x = a?
1) f(a) is defined, 2) lim (x → a) f(x) exists, 3) lim (x → a) f(x) = f(a).
p.5
First Principles of Derivatives
What is the definition of the derivative using First Principles?
f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h].
p.1
Derivative Calculation
What is the first step to find the derivative of the function f(x) = (1 + 2x^2)(3x^5)?
Use the product rule: f'(x) = u'v + uv', where u = (1 + 2x^2) and v = (3x^5).
Which of the following functions have a domain the set of all real numbers?
f(x) = e^(-x) + x, f(x) = 7sin(x), f(x) = 6/(x^2 - 2), f(x) = 5/(x + 1)
What does the Squeeze Theorem imply for the limit of (3 + cos(2x)) as x approaches infinity?
It shows that the limit is squeezed to 3.
p.5
First Principles of Derivatives
How do you apply the definition of the derivative to f(x) = 4x + 3?
Calculate f(x + h) = 4(x + h) + 3 and then use the limit definition.
p.1
Derivative Calculation
Given f(x) = g(h(x)), how do you find f'(10) if g(10) = 4, h(10) = 560, g'(10) = 0, and h'(10) = 35?
Use the chain rule: f'(10) = g'(h(10)) * h'(10) = g'(560) * 35.
p.6
Intermediate Value Theorem
What does the Intermediate Value Theorem guarantee for a continuous function on the interval [1, 5]?
There exists at least one c between 1 and 5 such that f(c) = 2.
At how many values of x is f discontinuous based on the provided graph?
The number of discontinuities (to be filled based on the graph).
p.2
Function Simplification
What is the first step to simplify tan(arccos(x))?
Use the identity tan(θ) = sin(θ)/cos(θ) and the relationship between sin and cos.
What is the limit of f(x) as x approaches 1 based on the provided graph?
The limit approaches a specific value (to be filled based on the graph).
What is the condition for δ in relation to x for the function f(x) = x/(x-1)?
If δ < |x - 1|, then |f(x) - 2| < 0.
Which of the following limits do NOT exist?
lim (x → 0) (3x^3 - 7)/(x^4 - 2x^3 + 3), lim (x → ∞) (x^2 - 10)/(2x^2 + 2)
What is the limit of f(x) as x approaches infinity based on the provided graph?
The limit approaches a specific value (to be filled based on the graph).
p.2
Intermediate Value Theorem
How can the Intermediate Value Theorem (IVT) be used to prove that x^2 + 1 = 0 has a root?
Show that f(-1) < 0 and f(0) > 0, indicating a root exists between -1 and 0.
p.1
Derivative Calculation
How do you find the derivative of the function f(x) = e^(x^2 + 3x)?
Use the chain rule: f'(x) = e^(x^2 + 3x) * (2x + 3).
p.6
Differentiability and Continuity
What can be said about the differentiability of the function based on the graph?
There are 2 values for which the function is not differentiable.
What conditions must a function f(x) satisfy if f'(x) = 0 for -2 < x < 2?
The function must be constant in that interval.
What is the limit as x approaches 1 for the expression |1 - x|/(x - 1)?
The limit does not exist as it approaches infinity.
What is the limit of f(x) as x approaches -6 based on the provided graph?
The limit approaches a specific value (to be filled based on the graph).
What is the significance of the limit conditions given for f(x) as x approaches -∞ and ∞?
They indicate the end behavior of the function.
p.6
Differentiability and Continuity
Which statements are true regarding the graphs of f(x) and g(x)?
g(x) is the derivative of f(x), g(x) is differentiable everywhere.
What does it mean for a function to be continuous on its domain?
There are no breaks, jumps, or holes in the function.
