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WRITTEN ASSIGNMENT 2 Hand in full solutions to the questions below. Make sure you justify all your work and include complete arguments and explanations. Your asnwers must be clear and neatly written, as well as legible (no tiny drawings or micro-handwriting please!). Your answers must be stapled, with your name and student number at the top of each page. 1. Come up with an expression for a function f defined for all real numbers but not continuous at x = 2 and such that lim f (x) = 3. x→2 2. Sketch the graph of a function f (x) whose domain is all real numbers except 0, and such that lim f (x) = 1, lim f (x) = 3, lim f (x) = ∞, lim f (x) = 0, and f (2) = 1. x→−2+ x→−2− x→0 x→1 3. Explain why the following functions are not continuous at x = 1. 2 (a) f (x) = 2 x −1 (b) ( 2 x +2x−3 if x = 6 1 x−1 g(x) = 4/3 if x = 1 (c) h(x) = ( 4x2 −4 2x+2 4 3x − 1 if if x>1 x≤1 4. Let f be a function continuous on [0, 1] except at 0.25 and such that f (0) = 1 and f (1) = 3. Is it true that for any such function f there exists a number c in (0, 1) such that f (c) = 2? Explain why or why not and draw one or more graphs to support your claim. 5. Using the theorem(s) discussed in class, show that the equation x4 + x = 3 has at least one solution. Make sure you justify your claim(s) and write a complete proof (use full sentences with punctuation). 6. Let f (x) = −2x2 + 6x. Show that the secant line through (2, f (2)) and (2 + h, f (2 + h)) for h 6= 0 has slope −2h − 2. Then use this information to compute the exact value of the slope of (a) the secant line through (2, f (2)) and (3, f (3)). (b) the tangent line to the graph of f at (2, f (2)). (c) Sketch the graph y = f (x), on the same diagram also draw the lines described in part (a) and (b). (d) Suppose f (x) describes the height of a moving object above the ground at some time t = x for 0 ≤ x ≤ 3. Explain what the slopes found in part (a) and (b) represent in this case. 1