Math 1131 Fall 2012 Review Problems for Exam 1 1. Solve the inequality: x2 + 3x − 4 > 0. 2. If f (x) = x2 − 2x + 3, evaluate the difference quotient f (a + h) − f (a) . h 3. Find the equation of a line that passes through the point (−2, 7) and is parallel to the line 2x + 3y = 6. 4. Let f (x) = x2 4 . Find a) f (−2) and b) the domain of f . −1 5. Find the range of the function h(x) = ln(x + 6). 6. Solve the following equations for x: (a) x2 + 6x − 8 = 0 √ (b) 2ln( 4x)−ln(2x + x) = 1 √ (c) e−x/2 · ex + 2 = 0 (d) cos2 x + cosx = 0 (e) ln x = 2 x (f) ee = 3 7. Rewrite the following expression as a single logarithm: 3ln 2−ln 8. Let f (x) = x2 − 1, let g(x) = √ x − 4, and let h(x) = (a) h−1 (x) (b) (f ◦ g)(x) (c) The domain of (g ◦ f )(x) 9. Find the inverse of the function f (x) = x+1 . 2x + 1 1 . 2 4 + 1. Find: x−1 10. If f (x) = 3x+ ln (x), find f −1 (3). 11. Use the table: x f (x) g(x) 1 2 3 4 5 6 3 2 1 0 1 2 6 5 2 3 4 6 to evaluate the expressions: (a) (f ◦ g)(3) (b) (g ◦ f )(1) (c) (g ◦ g)(6)  3  x − 4x − 6 if x > 1 −9 if x = 1 12. Let f (x) =  x+1 e if x < 1 (a) Is f (x) continuous at x = 1? Explain why or why not. (b) Explain why f (x) has a zero between x = 2 and x = 3.  √ if x < 0  −x −3 − x if 0 ≤ x < 3 13. Determine where g is discontinuous if g(x) =  (3 − x)2 if x > 3 14. Evaluate each of the following limits. Your answer should be in the form of a number, ∞, −∞, or ”does not exist” (DNE). x2 + x − 12 x→3 x2 − x − 6 2 x (b) lim √ sin 2 x→ π (a) lim (f) lim+ x→3 −2x + 1 x−3 (g) lim 9x2 + 7x + 3 x→5 3 3x − 5x + 2 x→∞ 4x4 + 2x2 + 6 (c) lim (h) lim x→0 (d) lim e2x x→−∞ 3x2 + x (e) lim 2 x→0 6x − x (i) lim x→2 x−1 + 5) x2 (x 2−x |2 − x| x . Find any vertical and horizontal asymptotes. Show, using limits, − 16 that these are asymptotes. 15. Let f (x) = x2 16. For the function f (x) = 3x + 2, we have limx→5 f (x) = 17. If we want to make |f (x) − 17| < 0.6, then how small do we have to make |x − 5|? Show all of your work. 17. Find the exact values of the expression: tan arcsin 1 2 . 18. The monthly cost of driving a car depends on the number of miles driven. Samantha found that in October it cost her $312.50 to drive 500 miles, and in February it cost her $375 to drive 1000 miles. Express the monthly cost C as a function of the distance driven, d, assuming that a linear relationship gives the suitable model. 19. Find the exponential function f (x) = Cax whose graph is given here: 20. The displacement (in meters) of an object moving in a straight line is given by s = t2 1 + 2t + , where t is measured in seconds. Find the average velocity over the time 4 period a) [1, 3], b) [1, 1.5], c) [1, 1.1], d) [1, 1.01]. Then estimate the instantaneous velocity at t = 1. 21. State the limit definition of the derivative. 22. Use the limit definition of the derivative to find f 0 (x) if f (x) = 2x2 − 5x + 1. 23. The graph of y = f (x) is given below. Sketch the graph of y = f 0 (x). 24. Using the limit definition of the derivative, find the equation of the tangent line to the curve y = f (x) = x2 − 1 at the point (2, 3). 25. Find the derivative of the following functions: √ (a) f (x) = 5 x − 4x − 4 1 3 (b) g(t) = t − 2t 2t − t 3 2x − 5 (c) h(x) = 2 x −x+1 x3 − 2x + 1 4 (d) f (x) = 3x − 2 x+5 26. Find the equation of the tangent line to the curve f (x) = x2 ex at x = 1. 27. A table of values for f , g, f 0 , and g 0 is given below: x f (x) g(x) f 0 (x) 1 3 2 4 2 1 8 5 3 7 2 7 (a) If h(x) = f (x)g(x), find h0 (2). f (x) (b) If h(x) = , find h0 (3). g(x) g 0 (x) 6 7 9