A1. The function is defined by (1)=1+= (a) Is J, even, odd, or neither? Justify you assertion. (b) Find the stationary points off and determine their nature. (c) Sketch the graph y= f(x), using your results from parts (a) and (b). (d) Explain why the Mean Value Theorem can be applied to / on [1/2, 2], and verify it by finding a suitable ce [1/2,2]. Give an interval where the Mean Value Theorem cannot be used for and justify your answer. A2. (a) The function : [0,00) [1,00) is defined by f(x)=√1+1². Show that is bijective. You must fully justify your answer. (b) The function g: RR is defined by g(x)=√1+r. Show that g is continuous at r = 0. Is g is differentiable at = 0? Justify your answer using the limit definition of the derivative. A3. (a) Starting with the definition tan(r) sin(r) cos(2)' find expressions for tan'(r) and tan" (z). (b) Calculate the Taylor polynomial of degree 2, centred at /4, for tan(2). (c) Using implicit differentiation, calculate arctan'(r). (Hint: use an expression for tan'(r) in terms of tan(r).) A4. Compute the following limits (a) lim tan(20) 0-10 tan(30) (b) lim (c) lim +123-72 +8r-3 A5. (a) The function / is defined by f(x) = 1- √√1+ Determine the integral F(x) = [* S(l) dt, by inspection, or using substitution. (b) Evaluate the improper integral Page 2 of 4 0 (Hint: write = 1/c in your result for F(z) in part (a).) Module Code: MATH105001 A6. Consider the function u(x,y) of two variables, given by u(x,y) =³+ (a) Compute the partial derivatives, y and zz, try and uyy- (b) Find the stationary points of the function u(x,y). Turn the page over B1. The function u is defined by u(x) = (1-2+) (a) Find an expression for us (sr) and ur" (s), and show that it" (r) = (1+2+2) (b) Show the u has a single stationary point, determine its nature, and then sketch the graph y= u(r). (c) If u(x) = f(x) g(x), then Leibniz's Rule states that ()(x)- n! k!(n-k)! (*)(x)g("-")(x). By choosing appropriate functions / and g, find an expression for ("). Then, show that ") (0) (n-1)(n-2) 2 (d) Hence, show that the Taylor Series for u, centred at 0, is u(x)=1+) n(n-3)! For what values of r does this series converge? B2. (a) Definite the hyperbolic functions cosh, sinh and tanh in terms of exponential functions. (b) Determine whether the functions cosh, sinh and tanh are odd, even or neither, using your definitions from part (a). Find lim tanh(r). Sketch the graphs of the three functions. 3-11:00 (c) Use your definition of sinh from part (a) to show that Page 3 of 4 Module Code: MATH105001 m²-1 sinh(logm) 2m Turn the page over (d) Use your definition of cosh from part (a) to show that cosh²(r) = cos(27) Hence, or otherwise, show that 1 log n cosh²(2) dr log(n) (n²-1)(n²+1) + 2 812 (e) Using similar techniques as in part (d), show that log cosh³(r)dr= (n-1)(n+10m²+1) 2413