Differentiation Questions
1. Show that the points (0, 0) and on the curve e(x + y) = cos (xy) have a common tangent.
(Total 7 marks)
2. The curve C has equation y = .
(a) Find the coordinates of the points on C at which = 0.
(4)
(b) The tangent to C at the point P(1, 2) cuts the x-axis at the point T. Determine the coordinates of T.
(4)
(c) The normal to C at the point P cuts the y-axis at the point N. Find the area of triangle PTN.
(7)
(Total 15 marks)
3. The function f is defined by f(x) = .
(a) Find f′(x).
(2)
(b) You are given that y = has a local minimum at x = a, a > 1. Find the value of a.
(6)
(Total 8 marks)
4. Find the equation of the normal to the curve x3y3 – xy = 0 at the point (1, 1).
(Total 7 marks)
5. Let f be a function defined by f(x) = x – arctan x, x .
(a) Find f(1) and f().
(2)
(b) Show that f(–x) = –f(x), for x .
(2)
(c) Show that x – , for x .
(2)
(d) Find expressions for f′(x) and f″(x). Hence describe the behaviour of the graph of f at the origin and justify your answer.
(8)
(e) Sketch a graph of f, showing clearly the asymptotes.
(3)
(f) Justify that the inverse of f is defined for all x and sketch its graph.
(3)
(Total 20 marks)
6. Let f be a function with domain that satisfies the conditions, f(x + y) = f(x) f(y), for all x and y and f (0) ≠ 0. (a) Show that f (0) = 1.
(3)
(b) Prove that f(x) ≠ 0, for all x .
(3)
(c) Assuming that f′(x) exists for all x , use the definition of derivative to show that f(x) satisfies the differential equation f′(x) = k f(x), where k = f′(0).
(4)
(d) Solve the differential equation to find an expression for f(x).
(4)
(Total 14 marks)
7. Consider the part of the curve 4x2 + y2 = 4 shown in the diagram below.
(a) Find an expression for in terms of x and y.
(3)
(b) Find the gradient of the tangent at the point .
(1)
(c) A bowl is formed by rotating this curve through 2π radians about the x-axis.
(Total 7 marks)
2. The curve C has equation y = .
(a) Find the coordinates of the points on C at which = 0.
(4)
(b) The tangent to C at the point P(1, 2) cuts the x-axis at the point T. Determine the coordinates of T.
(4)
(c) The normal to C at the point P cuts the y-axis at the point N. Find the area of triangle PTN.
(7)
(Total 15 marks)
3. The function f is defined by f(x) = .
(a) Find f′(x).
(2)
(b) You are given that y = has a local minimum at x = a, a > 1. Find the value of a.
(6)
(Total 8 marks)
4. Find the equation of the normal to the curve x3y3 – xy = 0 at the point (1, 1).
(Total 7 marks)
5. Let f be a function defined by f(x) = x – arctan x, x .
(a) Find f(1) and f().
(2)
(b) Show that f(–x) = –f(x), for x .
(2)
(c) Show that x – , for x .
(2)
(d) Find expressions for f′(x) and f″(x). Hence describe the behaviour of the graph of f at the origin and justify your answer.
(8)
(e) Sketch a graph of f, showing clearly the asymptotes.
(3)
(f) Justify that the inverse of f is defined for all x and sketch its graph.
(3)
(Total 20 marks)
6. Let f be a function with domain that satisfies the conditions, f(x + y) = f(x) f(y), for all x and y and f (0) ≠ 0. (a) Show that f (0) = 1.
(3)
(b) Prove that f(x) ≠ 0, for all x .
(3)
(c) Assuming that f′(x) exists for all x , use the definition of derivative to show that f(x) satisfies the differential equation f′(x) = k f(x), where k = f′(0).
(4)
(d) Solve the differential equation to find an expression for f(x).
(4)
(Total 14 marks)
7. Consider the part of the curve 4x2 + y2 = 4 shown in the diagram below.
(a) Find an expression for in terms of x and y.
(3)
(b) Find the gradient of the tangent at the point .
(1)
(c) A bowl is formed by rotating this curve through 2π radians about the x-axis.