SIM1002 Calculus I
Assignment
Answer ALL questions.
Q1. Differentiate the following:
(a) f(x) = |x− 5|, where x ≠ 5,
(b) esiny tan x = y3 +ln(cos x).
Q2. Prove that x/ln x ≤ e/1 for all values of ? > 0.
Q3. If ? = 3 sin bx − a cos 2x, where a and b are positive constants. Find the values of a and b if y + dx2/d2y = 6 cos 2x.
Q4. By using the Chain rule, prove the following statement.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.
Q5. Given x2 + 2xy + y2 = 1. Find dx/dy and dx2/d2y.
Q6. If F(x) = √1 + 5x. Find the coordinates of the point(s) on the graph of F where the normal line is parallel to the line 4x + 5Y = 1.
Q7. Use the Mean Value Theorem to show that sin x < x for 0 < x < 2π.
Q8. Given that a fixed point of a function f is a number ? in its domain such that f(a) = a. Use the Intermediate Value Theorem to show that any continuous function with domain [0,1] and range in [0,1] must have a fixed point.
Q9. Given f(x) = x2−x−2/x+2. Identify any local maximum and minimum point(s) of f(x) (if there is any). Find all the intervals of concavity and inflection point(s) of f(x) (if there is any).
Q10. State all asymptotes of the f(x) in Q9 and Sketch the graph of y = x2−x−2/x+2 by using the results from Q9.