MAT 1320B Final Exam [Apts)(11) We wish to evaluate J Van de. = 10 show that (a) By choosing an appropriate trigonometric substitution and simplifying the result, 2 4 do = / tan'(0) sec(0 ) do . (b) Evaluate / tan' (0) sec(0) do in terms of 0. (c) Express your answer to part (b) in terms of x according to the trigonometric substi tution you applied in part (a). Your final answer may not include any trigonometri functions or inverse trigonometric functions.MAT 1320B Final Exam 11 (12) We wish to find a numerical approximation (not the exact value!) for the integral using the midpoint rule. (a) Give the expression for this integral using a midpoint rule with n = 4. Write this expression as a sum that is ready to be evaluated, but do not evaluate it (that is, use all provided information but do not simplify the expression). 2 x* dx (b) Determine the smallest value of n that will guarantee that the approximation error using the midpoint rule for the integral / x4 dx is at most 0.01. Write a bound for this smallest n that is ready to be evaluated, but do not need to evaluate it.(10) Using the method of partial fractions (and not some other method), evaluate 12 + 3x-1 x(x2 + 1) (4pts] Be sure to give your final answer in terms of the original variable(5) A ladder of length 10 metres is leaning against the wall, making an angle 0 with the ground, The ladder slides down the wall, and the distance from the top of the ladder to the ground is decreasing at a rate of 0.1 meters per second at the moment when 0 = 4 radians, How fast is the angle 6 decreasing at that moment (in radians per second)? Give an exact answer (no need to simplify, though), not a decimal approximation, Be sure [3pts] to draw a diagram! 10m 0 = 71 0 Speed & radians/ second(7) Evaluate 17 In(t) dx 532 Cos? (t) dt. pts] (8) For each function f(x) below, give an antiderivative F(x). Your antiderivative should be defined on the entire domain of f. You do not need to include +C. . 2pts] For this question, no justification is required. (a) f(x) = I+1 F(x ) = (b) f() = sec(-x) tan(-2) F(x) = 1 (c) f(x) = F(x ) = (ac + 1 ) 2