Math 1A Practice FinalYou are allowed 1 sheet of notes. Calculators are not allowed. Each question is worth 3 marks, whichwill only be given for a clear and correct answer. There are questions on both sides of the paper.The questions on this practice final are all exercises in Stewart; the exercise number is given at the endof the question so you can check your answer.1. Draw the graph of y =√x + 3. (1.3.17)2. Prove that limx→0x2= 0 using the , δ definition of limit. (2.4.25)3. Prove that ex= 2 − x has at least one real root. (2.5.51)4. Differentiate ex/x2. (3.2.5)5. Find the derivative of tan(cos(x)). (3.5.29)6. Find dy/dx if x2y + xy2= 3x. (3.6.9)7. Find the absolute maximum and absolute minimum values of x/(x2+ 1) on [0,2]. (4.1.53)8. Prove that 2 s in−1x = cos−1(1 − 2x2) for 0 ≤ x ≤ 1. (4.2.32)9. Find limx→1+ln(x) tan(πx/2). (4.4.43)10. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle ofside L if one side of the rectangle lies on the base of the triangle. (4.7.21)11. Find the point on the line y = 4x + 7 that is closest to the origin. (4.7.15)12. Use Newton’s method to find 301/3to two decimal places. (4.9.11)13. Find the most general anti-derivative of 5x1/4− 7x3/4. (4.10.5)14. Find f given that f00(x) = 2 − 12x, f(0) = 9, f(2) = 15. (4.10.37)15. Estimate the area under the graph of f(x) = 1 + x2from x = −1 to x = 2 using three rectangles andright endpoints. (5.1.5a)16. Find an e xpression for the area under the graph of f(x) = x cos (x), 0 ≤ x ≤ π/2, as a limit. (5.1.19)17. Evaluate the integralR0−3(1 +√9 − x2)dx by interpreting it as an area. (5.2.37)18. Prove thatRπ /40sin3(x)dx ≤Rπ /40sin2(x)dx. (5.2.51)19. Find the derivative of g(x) =Rx0√1 + 2tdt. (5.3.7)20. Find the derivative of y =Rx3√x√t sin(t)dt. (5.3.51)21. Evaluate the integralR20(6x2− 4x + 5)dx. (5.4.17)22. Evaluate the integralRπ /40((1 + cos2(θ))/ cos2(θ))dθ. (5.4.33)123. Evaluate the indefinite integralR1+4x√1+x+2x2dx. (5.5.11)24. Evaluate the indefinite integralRcot(x)dx. (5.5.35)25. Evaluate the definite integralR20(x − 1)25dx. (5.5.49)26. Show that 1/2 + 1/3 + ··· + 1/n < ln(n). (5.6.3)27. Find the area enclosed by the curves x = 2y2, x + y = 1. (6.1.17)28. Find the volume of the region obtained by rotating y = x2, 0 ≤ x ≤ 2, y = 4, x = 0, about the y-axis.(6.2.5)29. Use the method of cylindrical shells to find the volume of a sphere of radius r. (6.3.43)30. Find the average value of (x − 3)2on [2, 5].
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