math 205 final exam winter 2023 for concordia university 11 main high score actual questions with calculated answers
math 205 final exam winter 2023
1 a sketch a graph of the function
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x x 0 0 MATH 205 FINAL EXAM WINTER 2023 FOR CONCORDIA UNIVERSITY 11 MAIN HIGH SCORE ACTUAL QUESTIONS WITH CALCULATED ANSWERS 1. (a) Sketch a graph of the function: 𝑓(𝑥) = {−1 + √4 − (𝑥 − 1)2, −1 ≤ 𝑥 ≤ 1 |𝑥 − 3| − 1 , 1 ≤ 𝑥 ≤ 4 4 on the interval [-1,4] and deduce the definite integral ∫–1 𝑓(𝑥)𝑑𝑥 in terms of the signed area (do not anti-differentiate). (b-1) Use the Fundamental Theorem of Calculus to find the derivative of the function 𝐹(𝑥) = 3x sin(t) 𝑑𝑡, then determine whether F is increasing or decreasing at x=2. ∫2x 2t+2 (b-2) Given that ∫2 𝑓(𝑡)𝑑𝑡 + 𝐴 = 𝑥2 + 𝑥 , find the value of A. Hint: Use the Fundamental Theorem of Calculus. (c-1) Consider the function 𝑓(𝑥) = 4 − 𝑥2. Write in sigma notation the formula for the left Riemann sum Ln on the interval [-1,2] using n subintervals of equal 2 length, and calculate ∫–1 𝑓(𝑥)𝑑𝑥 as the limit Ln as n➜∞. (c-2) Consider the function 𝑓(𝑥) = 4 − 𝑥2. Write in sigma notation the formula for the right Riemann sum Rn on the interval [ -1,2] using n subintervals of equal length, and approximate the area enclosed by f and the x -axis on the interval [-1,2] by calculating Rn with n=6. 2. Find the antiderivative 𝐹(𝑥) of the function 𝑓(𝑥) that satisfies the given condition: (a) ∫ 9x + 3x +3 𝑑𝑥 , 𝐹(0) = 0 (b) 3 x ∫ √16–x2 𝑑𝑥 , 𝐹(4) = 0 (c) ex ∫ 4+e2x 𝑑𝑥 , 𝐹(𝑙𝑛2) = π 8 3. Evaluate the following definite integrals (Give the exact value) : (a) ∫4 𝑥2√1 + 2𝑥𝑑𝑥 (b) ∫1 𝑥 sin–1(𝑥2) 𝑑𝑥 4. Evaluate the following indefinite integrals: (a) ∫ tan3(𝑥) sec5(𝑥) 𝑑𝑥 (b) ∫ √4 − 𝑥2𝑑𝑥
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