Topic 6: Calculus Differentiation Standard Level 6.1 Product Quotient Chain Rules Paper 1. Let f(x) = x 3 4x + 1. Expand (x + h) 3. Use the formula f (x) = lim h 0 f ( x + h) h f ( x) to show that the derivative of f(x) is 3x 4. The tangent to the curve of f at the point P(1, ) is parallel to the tangent at a point Q. Find the coordinates of Q. (d) The graph of f is decreasing for p < x < q. Find the value of p and of q. (e) Write down the range of values for the gradient of f. (Total 15 marks). Consider the curve with equation f(x) = px + qx, where p and q are constants. The point A(1, 3) lies on the curve. The tangent to the curve at A has gradient 8. Find the value of p and of q. (Total 7 marks) 3. The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x p)(x q), where p, q. Write down (i) the value of p and of q; the equation of the axis of symmetry of the curve. Find the equation of the function in the form y = (x h) + k, where h, k. Find d y. dx (d) Let T be the tangent to the curve at the point (0, 5). Find the equation of T. (Total 10 marks) 1
4. The equation of a curve may be written in the form y = a(x p)(x q). The curve intersects the x-axis at A(, 0) and B(4, 0). The curve of y = f (x) is shown in the diagram below. y 4 A B 4 0 4 6 x 4 6 (i) Write down the value of p and of q. Given that the point (6, 8) is on the curve, find the value of a. (iii) Write the equation of the curve in the form y = ax + bx + c. (i) Find d y. dx A tangent is drawn to the curve at a point P. The gradient of this tangent is 7. Find the coordinates of P. The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B. (i) Find the equation of L. Find the x-coordinate of the point where L intersects the curve again. (6) (Total 15 marks) 5. Let f (x) = x 3 x 1. Find f (x). Find the gradient of the curve of f (x) at the point (, 1). (Total 6 marks)
6. The main runway at Concordville airport is km long. An airplane, landing at Concordville, touches down at point T, and immediately starts to slow down. The point A is at the southern end of the runway. A marker is located at point P on the runway. A T P B km Not to scale As the airplane slows down, its distance, s, from A, is given by s = c + 100t 4t, where t is the time in seconds after touchdown, and c metres is the distance of T from A. The airplane touches down 800 m from A, (ie c = 800). (i) Find the distance travelled by the airplane in the first 5 seconds after touchdown. Write down an expression for the velocity of the airplane at time t seconds after touchdown, and hence find the velocity after 5 seconds. The airplane passes the marker at P with a velocity of 36 m s 1. Find (iii) how many seconds after touchdown it passes the marker; (iv) the distance from P to A. Show that if the airplane touches down before reaching the point P, it can stop before reaching the northern end, B, of the runway. (Total 15 marks) 7. A rock-climber slips off a rock-face and falls vertically. At first he falls freely, but after seconds a safety rope slows him down. The height h metres of the rock-climber after t seconds of the fall is given by: h = 50 5t, 0 t h = 90 40t + 5t, t 5 Find the height of the rock-climber when t =. Sketch a graph of h against t for 0 t 5. (1) 3
Find dh dt for: (i) 0 t t 5 (d) Find the velocity of the rock-climber when t =. (e) Find the times when the velocity of the rock-climber is zero. (f) Find the minimum height of the rock-climber for 0 t 5. (Total 15 marks) 8. Let f(x) = x cos x, for 0 x 6. Find f (x). Sketch the graph of y = f (x). (Total 7 marks) 9. Consider f(x) = x ln(4 x ), for < x <. The graph of f is given. Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the x-axis. (i) Find the x-coordinate of P and of Q. Consider f(x) = k. Write down all values of k for which there are exactly two solutions. Let g(x) = x 3 ln(4 x ), for < x <. x Show that g (x) = + 3x ln(4 x ). 4 x 4 Sketch the graph of g. (d) Consider g (x) = w. Write down all values of w for which there are exactly two solutions. (Total 14 marks) 4
10. The diagram below shows the graph of ƒ (x) = x e x for 0 x 6. There are points of inflexion at A and C and there is a maximum at B. Using the product rule for differentiation, find ƒ (x). Find the exact value of the y-coordinate of B. The second derivative of ƒ is ƒ (x) = (x 4x + ) e x. Use this result to find the exact value of the x-coordinate of C. (Total 6 marks) x 11. Consider the function h: x, x 1. ( x 1) A sketch of part of the graph of h is given below. y A P x Not to scale The line (AB) is a vertical asymptote. The point P is a point of inflexion. B Write down the equation of the vertical asymptote. Find h (x), writing your answer in the form a x ( x 1) n (1) where a and n are constants to be determined. Given that x 8 h ( x) =, calculate the coordinates of P. 4 ( x 1) (Total 8 marks) 5
1. Radian measure is used, where appropriate, throughout the question. 3x Consider the function y =. x 5 The graph of this function has a vertical and a horizontal asymptote. Write down the equation of (i) the vertical asymptote; the horizontal asymptote. d x Find, simplifying the answer as much as possible. d y How many points of inflexion does the graph of this function have? (1) (Total 6 marks) 13. Let f (x) = 1 + 3 cos (x) for 0 x π, and x is in radians. (i) Find f (x). Find the values for x for which f (x) = 0, giving your answers in terms of π. (6) The function g (x) is defined as g (x) = f (x) 1, 0 x π. (i) The graph of f may be transformed to the graph of g by a stretch in the x-direction with scale 1 factor followed by another transformation. Describe fully this other transformation. Find the solution to the equation g (x) = f (x) (Total 10 marks) 6
14. Note: Radians are used throughout this question. A mass is suspended from the ceiling on a spring. It is pulled down to point P and then released. It oscillates up and down. diagram not to scale P Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10 cos πt where t is the time in seconds from release. (i) What is the distance of the point P from the ceiling? How long is it until the mass is next at P? (i) Find d s. dt Where is the mass when the velocity is zero? (7) A second mass is suspended on another spring. Its distance r cm from the ceiling is modelled by the function r = 60 + 15 cos 4πt. The two masses are released at the same instant. (d) Find the value of t when they are first at the same distance below the ceiling. In the first three seconds, how many times are the two masses at the same height? (Total 16 marks) 7
15. The diagram shows the graph of the function f given by f (x) = A sin π x + B, for 0 x 5, where A and B are constants, and x is measured in radians. y (1,3) (5, 3) (0, 1) 0 1 3 4 5 x (3, 1) The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph. Write down the values of f (1) and f. Show that the period of f is 4. The point (3, 1) is a minimum point of the graph. Show that A =, and find the value of B. (d) Show that f (x) = π cos x The line y = k πx is a tangent line to the graph for 0 x 5. (e) Find (i) the point where this tangent meets the curve; the value of k. (f) Solve the equation f (x) = for 0 x 5. (6) (Total 4 marks) 8