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Math 132 - Test #1 Show all work except routine arithmetic computations. (10 pts) 1. The point P (9, 3) lies on the curve f(x) = this curve. √ √ x. Let Q be the point (x, x), also on (a) Write an expression that gives a general form for the slope of the secant line, P Q. (b) Compute the numerical value of this slope (to 4 decimal places) for each of x = 10, x = 9.5, x = 9.25 and x = 9.125. (These may be given as answers only.) (10 pts) 2. On the axes provided, sketch a clear picture of a single function satisfying all of these properties: (a) f(−1) = 1 (b) lim f(x) = 4 x→−1 (c) lim f(x) = 3 x→2− (d) lim+ f(x) = 1 x→2 (e) f is continuous on (−1, 2] (20 pts) 3. Show a step by step calculation of the following limits. You may combine any needed use of the Limit Laws into a single step. 3x2 − 2 x→−∞ 5x − 7x2 (a) lim (b) lim− x→3 |x − 3|x2 x−3 Math 132, Exam #1 2 (10 pts) 4. On the axes provided, sketch a graph of if x ≤ −1 2 f(x) = x2 if −1 < x ≤ 1 x − 1 if x > 1 (10 pts) 5. The ﬁgure shows the graph of a function named f with asymptotes marked as dashed lines. (a) Express, in correct limit language, all the asymptote relationships in the graph. (b) Write the equations of the aysmptotes. 3 -5 5 -3 x is shifted left by 4 units and then upward by 3 x2 + 1 units. What is an equation for the new, shifted graph? (10 pts) 6. Suppose that the graph of y = (10 pts) 7. Solve the equation 23x+1 = ex for x. Show your method. Give the answer both in exact form and as a decimal approximation. Math 132, Exam #1 3 (10 pts) 8. Consider the parametric equations of a curve; x = 2t − 1, y = 1 − 2t, −3 ≤ t ≤ 2. (a) Eliminate the parameter to ﬁnd the Cartesian equation of the curve. (b) Sketch the portion of the curve as described on the axes given. x2 + 2x − 3 . Showing all use of Limit Laws, x−1 evaluate lim f(x). Also, sketch a complete graph of y = (10 pts) 9. Let f(x) = x→1 f(x) on the accompanying axes.