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Questions and Answers
What is the nature of the predicate p ∧ p?
What is the nature of the predicate p ∧ p?
- Tautology (correct)
- Contradiction
- Dependent condition
- Neither
What can be concluded about the predicate p ∧ (¬ p)?
What can be concluded about the predicate p ∧ (¬ p)?
- It can be either true or false.
- It is always true if p is false.
- It is a tautology.
- It is a contradiction. (correct)
What is the result of the expression p ∨ (¬ p)?
What is the result of the expression p ∨ (¬ p)?
- It is a contradiction.
- It is always false.
- It is a tautology. (correct)
- It can be true or false.
What can be concluded about the limits of the function $f(x) = \sqrt{x^2 \cos(x)} + 3x^2$ as $x$ approaches 0 from the right?
What can be concluded about the limits of the function $f(x) = \sqrt{x^2 \cos(x)} + 3x^2$ as $x$ approaches 0 from the right?
Which statement about p → (¬ p) is correct?
Which statement about p → (¬ p) is correct?
Which of the following describes the conclusion about differentiability of the function $f(x) = x^4 - 2x^2$ at $x = 0$?
Which of the following describes the conclusion about differentiability of the function $f(x) = x^4 - 2x^2$ at $x = 0$?
If A = { k+k 1 , k ∈ N }, what can be determined about min A?
If A = { k+k 1 , k ∈ N }, what can be determined about min A?
If $mn$ is even, which of the following must be true?
If $mn$ is even, which of the following must be true?
When calculating the projection of vector u onto vector v, which of the following components is essential?
When calculating the projection of vector u onto vector v, which of the following components is essential?
What is the form of the parametric equations for a line that goes through points $P$ and $Q$?
What is the form of the parametric equations for a line that goes through points $P$ and $Q$?
What is the first step when defining the vectors u = PQ and v = PR from points P and Q?
What is the first step when defining the vectors u = PQ and v = PR from points P and Q?
In the case of sets defined by digits, how many odd numbers can be found in the set $A$, where each number consists of 4 digits from the set {1, 2, 3}?
In the case of sets defined by digits, how many odd numbers can be found in the set $A$, where each number consists of 4 digits from the set {1, 2, 3}?
What is the potential range of values for the expression k+k 1, where k belongs to the natural numbers?
What is the potential range of values for the expression k+k 1, where k belongs to the natural numbers?
For the function $f(x)$ defined as $f(x) = \sqrt{x^2 \cos(x)} + 3x^2$, what behavior should be observed as $x$ approaches 0 from the left?
For the function $f(x)$ defined as $f(x) = \sqrt{x^2 \cos(x)} + 3x^2$, what behavior should be observed as $x$ approaches 0 from the left?
What is the purpose of using mathematical induction in proving statements about $n$?
What is the purpose of using mathematical induction in proving statements about $n$?
What is the result of evaluating $lim_{x \to 0} \frac{f(x)}{x}$ for $f(x) = x^4 - 2x^2$?
What is the result of evaluating $lim_{x \to 0} \frac{f(x)}{x}$ for $f(x) = x^4 - 2x^2$?
What is the limit of $ rac{ ext{sin}(t) - 1}{t}$ as $t$ approaches infinity?
What is the limit of $ rac{ ext{sin}(t) - 1}{t}$ as $t$ approaches infinity?
What is the derivative $f'(x)$ for the function $f(x) = ext{sin}(x) ext{cos}(2x)$?
What is the derivative $f'(x)$ for the function $f(x) = ext{sin}(x) ext{cos}(2x)$?
What is the Taylor polynomial of order two for the function $f(x) = ext{sin}(x) ext{cos}(2x)$ around the point $x = rac{ ext{Ï€}}{2}$?
What is the Taylor polynomial of order two for the function $f(x) = ext{sin}(x) ext{cos}(2x)$ around the point $x = rac{ ext{Ï€}}{2}$?
At $x = rac{ ext{Ï€}}{2}$, how can you argue if $f(x) = ext{sin}(x) ext{cos}(2x)$ has a local minimum?
At $x = rac{ ext{Ï€}}{2}$, how can you argue if $f(x) = ext{sin}(x) ext{cos}(2x)$ has a local minimum?
Which of the following statements is TRUE regarding the function $f(x) = ext{sin}(x) ext{cos}(2x)$?
Which of the following statements is TRUE regarding the function $f(x) = ext{sin}(x) ext{cos}(2x)$?
What would happen to the limit of $ rac{ ext{sin}(t) - 1}{t}$ if you took the limit as $t$ approaches 0 instead?
What would happen to the limit of $ rac{ ext{sin}(t) - 1}{t}$ if you took the limit as $t$ approaches 0 instead?
If $f''(x)$ is negative at $x = rac{ ext{Ï€}}{2}$ for the function $f(x) = ext{sin}(x) ext{cos}(2x)$, what does that indicate?
If $f''(x)$ is negative at $x = rac{ ext{Ï€}}{2}$ for the function $f(x) = ext{sin}(x) ext{cos}(2x)$, what does that indicate?
In computing the Taylor polynomial, what is a critical first step?
In computing the Taylor polynomial, what is a critical first step?
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Study Notes
Introduction to Mathematics & Calculus 1A - Sample Test 1
- This exam has 11 exercises.
- Exercises 1-4 require only final answers to be filled on the answer form.
- Exercises 5-11 require full calculations and argumentation to be written on the answer form.
- Electronic devices are not allowed.
Sets
- A = {k + k^1, k ∈ N}
- Find the infimum (inf A), maximum (max A), minimum (min A), and supremum (sup A) of the set A.
Logic
- Determine whether the following predicates are a tautology, a contradiction, or neither:
- p ∧ p
- p ∧ (¬ p)
- p ∨ (¬ p)
- p → (¬ p)
Vectors
- P = (1, 1, 0)
- Q = (1, 1, 1)
- R = (2, -1, 2)
- u = PQ (vector from P to Q)
- v = PR (vector from P to R)
- Calculate the cross product of u and v (u × v).
- Calculate the projection of u onto vector v.
Lines and Planes in R3
- P = (1, -1, 4)
- Q = (3, 2, 3)
- Find the parametric equations for the line â„“1 passing through points P and Q.
- Find the equation of the plane that passes through point P and is perpendicular to line â„“2:
- x = 1 - t
- y = 3 - 2t
- z = 2 - t
- t ∈ R
- Simplify the equation of the plane as much as possible.
Mathematical Induction
- Prove that for all n ∈ N ∪ {0}, the following holds:
- ∑(i=0 to n) (i+2)/(n+3) = n/(n+3)
- Use the hint: (n+r 1) = (r−n 1) + (nr) for all n, r ∈ N, r ≤ n.
Sets and Counting
- A is a set of 4-digit numbers where each digit is from the set {1, 2, 3}.
- Determine the number of odd numbers in set A.
- Determine the number of numbers in A that are odd or start with digit 1.
Limits and Continuity
- f(x) = √(x^2 * cos(x) + 3x^2)
- Calculate the limit of f(x)/x as x approaches 0 from the right (lim x→0+ f(x)/x) and from the left (lim x→0− f(x)/x)
- Conclude that lim x→0 f(x)/x does not exist.
- Show that f is not differentiable at x = 0.
Derivatives and Critical Points
- f(x) = x^4 - 2x^2
- Find all critical points of f.
- Find the absolute minimum and maximum values of f on the interval [-1, 2].
Limits at Infinity
- Compute the limit as t approaches infinity of (sin(t) - 1) / t or argue that the limit does not exist.
Taylor Polynomials
- f(x) = sin(x) * cos(2x)
- Show that f'(x) = cos(x) * cos(2x) - 2 * sin(x) * sin(2x).
- Compute the Taylor polynomial of order 2 around the point x = π/2.
- Argue that f has a local minimum at x = π/2.
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