Introduction to Mathematics & Calculus 1A Sample Test 1 PDF

Course : Introduction to Mathematics & Calculus 1A Date : Module Course code : : 1A Time : 3 hours Reference :...

Course : Introduction to Mathematics & Calculus 1A Date : Module Course code : : 1A Time : 3 hours Reference : A Introduction to Mathematics & Calculus 1A Sample Test 1 Instructions This exam contains 11 exercises. You shall use the provided answer form to submit your answers. ▶ For exercises 1–4, you are only required to fill in the final answer on the answer form. ▶ For exercises 5–11, you are required to write down a full calculation and argumentation. You will hand in your answer form only. Any text outside the answer form will not be considered. If you run out of space, you can use the extra space at the end of the answer form. Refer clearly to that space in the original answer. Do not write with red pen or pencil. Do not use correction fluid or tape. Á THE USE OF ELECTRONIC DEVICES IS NOT ALLOWED. Á Final answer questions Report all your answers on the answer form. 1. Let A = { k+k 1 , k ∈ N }. [2 pt] (a) Determine inf A. (b) Determine max A (c) Determine min A (d) Determine sup A. 2. These are multiple choice final answer questions. Per question, only one multiple choice answer is [2 pt] correct. 4 correct answers: 2 pt 3 correct answers: 1 pt 2 or less correct answers: 0 pt Provide your final answer (and only your final answer) on the answer sheet. For each of the following predicates, determine whether it is a tautology, a contradiction or neither: (a) p ∧ p (b) p ∧ (¬ p) (c) p ∨ (¬ p) (d) p → (¬ p) −→ −→ 3. Let P = (1, 1, 0), Q = (1, 1, 1), and R = (2, −1, 2). Define the vectors u = PQ and v = PR. (a) Calculate u × v. [1 pt] (b) Calculate the projection of u onto v. [2 pt] Unauthorized reproduction or distribution is forbidden. 4. Let P = (1, −1, 4) and let Q = (3, 2, 3) be two points in R3. (a) Find parametric equations for the line ℓ1 through P and Q. [2 pt] (b) Find an equation of the plane through the point P which is perpendicular to the line [2 pt] ℓ2 : x = 1 − t, y = 3 − 2t, z = 2 − t, t ∈ R. Simplify the equation as much as possible. Open questions The full solutions to exercises 5–11 must be clearly written down on the answer form, including calculations and argumentations. Points will not be awarded for reaching a correct result if this is not supported by a correct procedure and by a sound and clear argumentation. 5. Let m, n ∈ Z. Either prove or give a counterexample to the following statement: [3 pt] If mn is even, then m is even, or n is even (or both). 6. Use mathematical induction on n to prove that for all n ∈ N ∪ {0}, [4 pt] n ( ) ( ) i+2 n+3 ∑ i = n. i =0 Hint: Use that (n+r 1) = (r−n 1) + (nr) for all n, r ∈ N, r ≤ n. 7. Consider the set A of numbers consisting of 4 digits where each digit is from the set {1, 2, 3}. For example, 1311 ∈ A. (a) How many odd numbers are in A? [1 pt] (b) How many numbers in A are odd, or start with the digit 1 (or both)? [2 pt] 8. Define the function f : R → R as follows: √ f (x) = x2 cos( x ) + 3x2. f (x) f (x) f (x) (a) Calculate limx→0+ x and limx→0− x , and conclude that limx→0 x does not exist. [3 pt] (b) Show that f is not differentiable at x = 0. [2 pt] 9. Define the function f : R → R as follows: f ( x ) = x4 − 2x2. (a) Find all critical points of f. [2 pt] (b) Find the absolute extreme values of f on the interval [−1, 2], and indicate whether they [2 pt] are an absolute minimum or an absolute maximum. Continues on the following page.  10. Compute the limit [2 pt] sin(t) − 1 lim t→∞ t or argue that the limit does not exist. 11. Define the function f : R → R as follows: f ( x ) = sin( x ) cos(2x ). (a) Show that f ′ ( x ) = cos( x ) cos(2x ) − 2 sin( x ) sin(2x ). [1 pt] π (b) Compute the Taylor polynomial of order two around the point x = 2. [2 pt] π (c) Argue that f has a local minimum at x = 2. [1 pt] Total: 36 pt Unauthorized reproduction or distribution is forbidden.  Σ A 1 2 3 4 5 6 7 8 9 10 11 Only for sorting. Do not fill in this table. Answer form Introduction to Mathematics & Calculus 1A – Sample Test 1 – Full name in BLOCK LETTERS Student number Programme Á FILL IN YOUR DATA AS SOON AS YOU RECEIVE THE EXAM Á Question 1. A B C D Question 2. Fill in the correct answers: (a) Tautology Contradiction Neither (b) Tautology Contradiction Neither (c) Tautology Contradiction Neither (d) Tautology Contradiction Neither Question 3. (a) (b) Question 4. (a) (b) 1 Question 5. Write a full calculation/argumentation clearly in the box. 2 Question 6. Write a full calculation/argumentation clearly in the box. 3 Question 7a. Write a full calculation/argumentation clearly in the box. Question 7b. Write a full calculation/argumentation clearly in the box. Question 8a. Write a full calculation/argumentation clearly in the box. Question 8b. Write a full calculation/argumentation clearly in the box. 4 Question 9a. Write a full calculation/argumentation clearly in the box. Question 9b. Write a full calculation/argumentation clearly in the box. Question 10. Write a full calculation/argumentation clearly in the box. Question 11a. Write a full calculation/argumentation clearly in the box. 5 Question 11b. Write a full calculation/argumentation clearly in the box. Question 11c. Write a full calculation/argumentation clearly in the box. Additional writing space. Clearly refer to this space in the original answer 6 Additional writing space. Clearly refer to this space in the original answer 7 Additional writing space. Clearly refer to this space in the original answer 8