Math 1A Practice Final You are allowed 1 sheet of notes Calculators are not allowed Each question is worth 3 marks which will 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 end of the question so you can check your answer 1 Draw the graph of y x 3 1 3 17 2 Prove that limx 0 x2 0 using the 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 x2 y xy 2 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 sin 1 x cos 1 1 2x2 for 0 x 1 4 2 32 9 Find limx 1 ln x tan x 2 4 4 43 10 Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side 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 3 to 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 f 00 x 2 12x f 0 9 f 2 15 4 10 37 15 Estimate the area under the graph of f x 1 x2 from x 1 to x 2 using three rectangles and right endpoints 5 1 5a 16 Find an expression for the area under the graph of f x x cos x 0 x 2 as a limit 5 1 19 17 Evaluate the integral 18 Prove that R 4 0 R0 3 1 sin3 x dx 21 Evaluate the integral R2 22 Evaluate the integral R 4 0 0 9 x2 dx by interpreting it as an area 5 2 37 R 4 19 Find the derivative of g x 20 Find the derivative of y 0 Rx 0 R x3 sin2 x dx 5 2 51 x 1 2tdt 5 3 7 t sin t dt 5 3 51 6x2 4x 5 dx 5 4 17 1 cos2 cos2 d 5 4 33 1 23 Evaluate the indefinite integral R 1 4x dx 1 x 2x2 24 Evaluate the indefinite integral R cot x dx 5 5 35 25 Evaluate the definite integral R2 0 5 5 11 x 1 25 dx 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 2y 2 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 2 on 2 5 6 5 9a 2
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