Tutorial: Task 1: Consider signal $f(t)$ in Figure 1. Find the signal $f(2t-3)$ and demonstrate it graphically. 1 -2 -1 0 Figure 1 Task 2: Are the following systems linear? Why? y(t) = x(t) y(t) = cos(x(t)) Task 3: Are the following systems time-invariant? Why? y(t) = tx(t) y(t) = \frac{x(t)}{x(t)} y(t) = x(t-2) + x(2-t)
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2.2 For each of the following systems, specify whether or not the system is: (i) linear and/or (ii) time-invariant. (a) y(t) = 3x(t - 1) (b) y(t) = t x(t) (c) dy/dt + y(t - 1) = x(t) (d) dy/dt + 2y(t) = ∫(from -∞ to t) x(τ) dτ (e) y(t) = x(t) u(t) (f) y(t) = ∫(from t to ∞) x(τ) dτ (g) y(t) = ∫(from t to 2t) x(τ) dτ
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7. (15 pts) Pyridine undergoes nitration under vigorous conditions to furnish a single mononitro derivative as shown. Your task will be to predict which isomer is formed and justify your answer. (a) (9 pts) Under these highly acidic conditions, the species undergoing nitration is the conjugate acid of pyridine. Write the important contributing resonance structures for the intermediates formed by attack of NO2+ at the 2-, 3-, and 4-positions of the conjugate acid of pyridine. Note: There are three contributing resonance structures for attack at each position. (b) (3 pts) For each set of resonance structures in part (a), indicate which, if any, are particularly important contributors to the stability of the intermediate cation and explain your reasoning. (c) (3 pts) Draw the structure of the expected product and explain why you expect it to be preferred.
Text: 1 1 grrh ot 3 1 C 4 LEI 8 duund 103j47 JLE Jpil 4 8 F 9 ano / D M ne [oll
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