Unit 1 - Exam Preparation Flashcards
What is the limit of f(x) as x approaches a?
The value that f(x) approaches as x gets arbitrarily close to a.
True or False: The limit of a function can exist even if the function is not defined at that point.
True
Fill in the blank: The notation for the limit of f(x) as x approaches a is ______.
lim(x->a) f(x)
What is the definition of a derivative?
The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero.
What is the derivative of f(x) = x^2?
f’(x) = 2x
Multiple Choice: What is the derivative of sin(x)? A) cos(x) B) -cos(x) C) sin(x) D) -sin(x)
A) cos(x)
True or False: The derivative of a constant function is always zero.
True
What is the power rule for differentiation?
If f(x) = x^n, then f’(x) = n*x^(n-1).
Multiple Choice: Which of the following is not a rule for finding derivatives? A) Product Rule B) Quotient Rule C) Limit Rule D) Chain Rule
C) Limit Rule
Fill in the blank: The ______ states that if f is continuous on [a, b] and f(a) and f(b) have opposite signs, then there exists at least one c in (a, b) such that f(c) = 0.
Intermediate Value Theorem
What does the Mean Value Theorem state?
If a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f’(c) = (f(b) - f(a)) / (b - a).
True or False: A function can have a derivative at a point where it is not continuous.
False
What is the limit of f(x) = 1/x as x approaches 0 from the right?
Infinity
Multiple Choice: What is the derivative of e^x? A) e^x B) x*e^x C) ln(x) D) 1/x
A) e^x
Fill in the blank: The ______ is used to find the slope of the tangent line to a curve at a given point.
Derivative
What is the definition of continuity at a point?
A function f is continuous at a point a if lim(x->a) f(x) = f(a).
True or False: If the limit of f(x) as x approaches a is equal to f(a), then f is continuous at a.
True
What is the derivative of ln(x)?
1/x
Multiple Choice: Which of the following represents the Chain Rule? A) (f*g)’ = f’g B) (f(g(x)))’ = f’(g(x)) * g’(x) C) (f/g)’ = (f’g - fg’)/g^2 D) None of the above
B) (f(g(x)))’ = f’(g(x)) * g’(x)
Fill in the blank: The ______ of a function represents its instantaneous rate of change.
Derivative
What is the limit of f(x) = x^2 as x approaches 3?
9
True or False: The derivative of a function provides information about the function’s increasing and decreasing behavior.
True
What is the first derivative test used for?
To determine local maxima and minima of a function.
Fill in the blank: A function is ______ if its derivative is positive over an interval.
Increasing
What is the second derivative test used for?
To determine the concavity of a function and to identify points of inflection.
Multiple Choice: Which of the following is the derivative of x^3? A) 3x^2 B) 2x^2 C) x^2 D) 3x
A) 3x^2
What is the limit of f(x) = 1/x as x approaches infinity?
0
True or False: A function can have a horizontal asymptote if its limit approaches a finite number as x approaches infinity.
True
Fill in the blank: The ______ of a function describes its end behavior as x approaches infinity or negative infinity.
Asymptote
What is the derivative of cos(x)?
-sin(x)
True or False: The product rule states that the derivative of a product of two functions is the product of their derivatives.
False
What is the formula for the product rule?
(f*g)’ = f’g + fg’
Multiple Choice: What is the limit of sin(x)/x as x approaches 0? A) 0 B) 1 C) Infinity D) Does not exist
B) 1
Fill in the blank: The ______ states that for every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - L| < ε.
Epsilon-Delta Definition of a Limit
What is the derivative of tan(x)?
sec^2(x)
True or False: The quotient rule states that the derivative of a quotient of two functions is the quotient of their derivatives.
False
What is the formula for the quotient rule?
(f/g)’ = (f’g - fg’)/g^2
What is the limit of f(x) = x^2 - 4 as x approaches 2?
0
Fill in the blank: A function is ______ if its derivative is negative over an interval.
Decreasing
What is the limit of f(x) = sqrt(x) as x approaches 4?
2
True or False: The derivative of a function can be interpreted as the slope of the tangent line at any point on the function.
True
What is the derivative of a constant multiplied by a function?
The constant multiplied by the derivative of the function.
Fill in the blank: The ______ of a function is used to find where the function changes from increasing to decreasing or vice versa.
Critical Point
What is the limit of f(x) = 1/(x^2) as x approaches infinity?
0
Multiple Choice: What is the derivative of x^4? A) 4x^3 B) 3x^4 C) x^3 D) 4x^2
A) 4x^3
What is the limit of f(x) = 2x + 3 as x approaches 1?
5
True or False: The derivative provides no information about the concavity of a function.
True
What does the term ‘differentiable’ mean?
A function is differentiable at a point if it has a derivative at that point.
Fill in the blank: A point where a function’s derivative does not exist is called a ______.
Point of Non-Differentiability
What is the limit of f(x) = tan(x) as x approaches pi/2?
Infinity
True or False: A function can be continuous but not differentiable.
True
What is the derivative of a sum of two functions?
The sum of the derivatives of the two functions.
Fill in the blank: The ______ of a function can be used to analyze its behavior, such as increasing, decreasing, and concavity.
Derivative
What is the limit of f(x) = 3x^2 + 2 as x approaches 0?
2
Multiple Choice: Which of the following is the derivative of 5x? A) 5 B) 5x^2 C) 1 D) 0
A) 5
What is the limit of f(x) = 2/x as x approaches 0 from the left?
-Infinity
True or False: The limit of a function must be unique.
True
What is the derivative of a function evaluated at a point?
The slope of the tangent line to the function at that point.
Fill in the blank: The ______ of a function can indicate the presence of local extrema.
First Derivative
What is the limit of f(x) = sin(x)/x as x approaches 0?
1
Multiple Choice: What is the derivative of 7x^5? A) 35x^4 B) 7x^4 C) 35x^5 D) 5x^6
A) 35x^4
What is the limit of f(x) = ln(x) as x approaches 0 from the right?
-Infinity
True or False: The second derivative can provide information about the concavity of a function.
True
What is the derivative of a function at a point defined as?
The limit of the difference quotient as the interval approaches zero.
Fill in the blank: The ______ of a function is a graphical representation of its rate of change.
Derivative
What is the limit of f(x) = x^2 - 1 as x approaches 1?
0
Definition of a function:
A function y = f (x) is a set of points (x, y) such that for each x there is one and only one y. The function’s domain is the set of all x-values in the set of points. The function’s range is the set of all y-values in the set of points.
Definition of a Limit:
Let f be a function defined at all values in an open interval containing a number a, with the possible exception of a itself. If values of f (x) approach a number L whenever values of x 6 = a approach the number a, then we say that the limit of f (x) as x approaches a is L
The Squeeze Theorem:
If f (x) ≤ g(x) ≤ h(x) for all x in an open interval containing a, except
possibly at a itself, and if lim_x->a f (x) and lim_x->a h(x) both exist and are equal to L, then lim_x->a g(x) exists and is also equal to L
A function is continuous when:
- f (a) is defined
- lim_x->a f (x) exists
- lim_x-> f (x) = f (a)
Removable Discontinuity:
f has a limit at a but either this limit does not equal f (a) or f (a) does not exist
Jump Discontinuity:
f has one-sided limits at a but they are not equal. f (a) may or may not exist
Infinite Discontinuity:
one or both one-sided limits of f at a don’t exist. f (a) may or may not exist
A continuous function is….
a function that is continuous on its domain
Properties of Continuity:
- c f where c is a number
- f ± g
- f ⋅ g
- f/g, for g (a) ≠ 0
