Question The diagram shows the part of the curve y = xe-x for , and its maximum point M. i. Find the x-coordinate of M. ii. Use the trapezium rule with two intervals to estimate the value of giving your answer correct to 2 decimal places. iii. […]
Question i. Express in the form , where and , giving the exact values of R and . ii. Hence show that iii. By differentiating , show that if then . iv. Using the results of parts (ii) and (iii), show that Solution i. We […]
Question The diagram shows the curve y = 2ex + 3e-2x. The curve cuts the y-axis at A. i. Write down the coordinates of A. ii. Find the equation of the tangent to the curve at A, and state the coordinates of the point where this tangent meets the x-axis. […]
Question The curve with equation y = x2 ln x, where x > 0, has one stationary point. i. Find the x-coordinate of this point, giving your answer in terms of e. ii. Determine whether this point is a maximum or a minimum point. Solution i. We […]
Question The parametric equations of a curve are Where t takes all positive values. i. Show that ii. Find the equation of the tangent to the curve at the point where . iii. The curve has one stationary point. Find the y-coordinate of this point, and determine whether […]
Question A curve has equation . i. Write down expressions for and ii. Find the coordinates of the stationary point on the curve and determine its nature. iii. Find the volume of the solid formed when the region enclosed by the curve, the x-axis and the lines x = 1 and x = 2 is rotated […]
Question A curve is such that and is a point on the curve. i. Find the equation of the normal to the curve at P, giving your answer in the form . ii. Find the equation of the curve. Solution i. To find the equation of the normal to the curve at P; To find the equation […]
Question The equation of a curve is and the equation of a line is . The curve and the line intersect at the points A and B. i. The mid-point of AB is M. Show that the coordinates of M are . ii. Find the coordinates of the point Q on the curve at which the tangent is parallel […]
Question The diagram shows a glass window consisting of a rectangle of height m and width m and a semicircle of radius m. The perimeter of the window is 8 m. i. Express in terms of . ii. Show that the area of the window, A m2, is given by Given that can vary, iii. find the value […]
Question The diagram shows part of the graph of and the normal to the curve at . This normal meets the -axis at R. The point Q on the -axis and the point S on the curve are such that PQ and SR are parallel to the -axis. i. Find the equation of the normal at P […]