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Questions and Answers
What is the nature of the predicate p ∧ p?
What can be concluded about the predicate p ∧ (¬ p)?
What is the result of the expression p ∨ (¬ p)?
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?
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Which statement about p → (¬ p) is correct?
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Which of the following describes the conclusion about differentiability of the function $f(x) = x^4 - 2x^2$ at $x = 0$?
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If A = { k+k 1 , k ∈ N }, what can be determined about min A?
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If $mn$ is even, which of the following must be true?
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When calculating the projection of vector u onto vector v, which of the following components is essential?
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What is the form of the parametric equations for a line that goes through points $P$ and $Q$?
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What is the first step when defining the vectors u = PQ and v = PR from points P and Q?
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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}?
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What is the potential range of values for the expression k+k 1, where k belongs to the natural numbers?
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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?
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What is the purpose of using mathematical induction in proving statements about $n$?
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What is the result of evaluating $lim_{x \to 0} \frac{f(x)}{x}$ for $f(x) = x^4 - 2x^2$?
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What is the limit of $ rac{ ext{sin}(t) - 1}{t}$ as $t$ approaches infinity?
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What is the derivative $f'(x)$ for the function $f(x) = ext{sin}(x) ext{cos}(2x)$?
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What is the Taylor polynomial of order two for the function $f(x) = ext{sin}(x) ext{cos}(2x)$ around the point $x = rac{ ext{π}}{2}$?
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At $x = rac{ ext{π}}{2}$, how can you argue if $f(x) = ext{sin}(x) ext{cos}(2x)$ has a local minimum?
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Which of the following statements is TRUE regarding the function $f(x) = ext{sin}(x) ext{cos}(2x)$?
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What would happen to the limit of $ rac{ ext{sin}(t) - 1}{t}$ if you took the limit as $t$ approaches 0 instead?
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If $f''(x)$ is negative at $x = rac{ ext{π}}{2}$ for the function $f(x) = ext{sin}(x) ext{cos}(2x)$, what does that indicate?
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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 ℓ1 passing through points P and Q.
- Find the equation of the plane that passes through point P and is perpendicular to line ℓ2:
- 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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Description
This sample test covers various topics in Mathematics and Calculus 1A, including sets, logic, vectors, and lines and planes in R3. It consists of 11 exercises requiring both final answers and detailed calculations. No electronic devices are allowed during the exam.
