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MATH1151 Revision Questions+Solutions

MATH1151 Revision Questions+Solutions - MATH1151 Student...

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MATH1151 Student Support Revision Questions 1. (a) Z ln 3 0 x sinh x dx (b) Show that cosh 2 x - sinh 2 x = 1 and deduce that tanh x is an increasing function. 2. Determine each of the following limits. Formally prove your answers to (a) and (b). (a) lim x 1 + 1 x - 1 (b) lim x →∞ x 2 + 1 x 2 (c) lim x 0 x log(1 + x ) ( e x - 1) 2 (d) lim x 1 Z x 1 cos( t 2 ) dt sin( πx ) (e) lim x 2 + x 2 - x - 2 x - 2 3. Suppose that f ( x ) = 1 2 ( x + 1) for - 1 x < 1 , x 2 - 4 x + 4 for 1 x < 2 , - x 2 + 4 x - 4 for 2 x < 3 , f ( x + 4) for all real x. Is f continuous at x = 1? Is f continuous at x = - 1? Is f differentiable at x = 2? Give reasons. 4. Sketch the polar curve r = | sin θ | where 0 θ 2 π and determine the cartesian equation(s) of the curve. 5. (a) Divide the interval 1 x 2 into n equal parts, and show that n X j =1 1 n + j < Z 2 1 1 x dx < n - 1 X j =0 1 n + j . (b) Find the difference between the upper and lower sums. Use this to show that the error in the approximation ln 2 10 5 X j =1 1 10 5 + j is less than 5 × 10 - 6 . 6. (a) Show that Z 0 1 x + x 2 + 1 φ dx = 1, where φ = 1 + 5 2 . (b) Prove that the improper integral Z 1 x 1 + 5 x 2 dx converges and give a numerical upper bound for its value. (c) Evaluate the improper integral Z 2 1 2 x - 3 x 2 - 1 dx after first writing it explicitly as a limit. 7. Show that e x 1 + x for all real numbers x . Then, using the fact that e x = 1 + Z x 0 e t dt , show that e x 1 + x + x 2 2 for all positive real numbers x . 8. (a) Suppose a > - 1 and b > - 1. Use Leibniz’s rule to show that Z 1 0 x a - 1 ln x dx = ln( a + 1). Hence, or otherwise, evaluate Z 1 0 x a - x b ln x dx (b) Let x ( t ) = 1 2 Z t 0 sinh[2( t - s )] f ( s ) ds where f is a continuous function. Show that d 2 x dt 2 - 4 x = f ( t ). 9. The error term involved in using Simpson’s Rule can be written as Ef = - f (4) ( c )( b - a ) 5 2880 n 4 where c ( a, b ). Estimate the number of subintervals needed to evaluate Z 3 1 dx x 3 with Simpson’s Rule and error less than 10 - 6 . 10. Suppose w = x + sin( yz ) and x = 2 s + 1 , y = s - t, z = te s . Determine ∂w ∂s at ( s, t ) = (0 , 1). Version 1.2 1
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MATH1151 Student Support Revision Questions 11. Consider the plane π that passes through the points A (1 , 0 , - 1) T , B (3 , - 1 , 0) T and C (2 , 1 , 1) T . Find the angle ABC , the area of the triangle ABC , the equation of π in parametric vector, cartesian and point-normal forms, and the point(s) of intersection, if any, of π with the line x - 1 3 = y + 2 1 = z 6 . 12. Find the shortest distance between: (a) the point (8 , 1 , - 5) T and the line ˜ x = (1 , 2 , - 1) T + λ (3 , 1 , 0) T ; (b) the point (6 , - 5 , 8) T and the plane x - y + 3 z = 2; (c) the lines ˜ x = (1 , - 3 , 9) T + λ (3 , 2 , 5) T and ˜ y = (5 , 0 , 4) T + μ (2 , 1 , 3) T . 13. Use the method of least squares to determine the line of best fit: ( - 2 , 3), (1 , - 2), (4 , - 7), (5 , 2). 14. Show that the matrix A = 1 2 5 3 2 - 1 2 1 0 is invertible, and find A - 1 . Verify that AA - 1 = I and A - 1 A = I . 15. Determine the values of a and b such that the system of linear equations: - x + 4 y = - 5 and 3 x + ay = b has (a) infinitely many solutions; (b) no solution; (c) a unique solution.
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