Give exact answers in this part of the question.
The temperature g (t) at time t of a given point of a heated iron rod is given by
g (t) = , where t > 0.
(a) Find the interval where g¢ (t) > 0.
(b) Find the interval where g² (t) > 0 and the interval where g² (t) < 0.
(c) Find the value of t where the graph of g (t) has a point of inflexion.
(d) Let t* be a value of t for which g¢ (t*) = 0 and g² (t*) < 0. Find t*.
(e) Find the point where the normal to the graph of g (t) at the point
(Total 18 marks)
Let f (x) = ln |x5 – 3x2|, –0.5 < x < 2, x ¹ a, x ¹ b; (a, b are values of x for which f (x) is not defined).
(a) (i) Sketch the graph of f (x), indicating on your sketch the number of zeros of f (x). Show also the position of any asymptotes.
(ii) Find all the zeros of f (x), (that is, solve f (x) = 0).
(b) Find the exact values of a and b.
(c) Find f (x), and indicate clearly where f¢ (x) is not defined.
(d) Find the exact value of the x-coordinate of the local maximum of f (x), for 0 < x < 1.5. (You may assume that there is no point of inflexion.)
(e) Write down the definite integral that represents the area of the region enclosed by f (x) and the x-axis. (Do not evaluate the integral.)
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