Question The equation of a curve is i. Show that ii. Find the coordinates of each of the points on the curve where the tangent is parallel to the x- axis. Solution i. We are given; We are required to find . To find from an implicit equation, […]
Question The curve with equation has one stationary point in the interval . Find the exact x-coordinate of this point. Solution We are required to find the x-coordinate of stationary point of the curve with equation; A stationary point on the curve is the point where gradient of the curve is equal to […]
Question i. By differentiating , show that if θ then . ii. Hence show that Giving the values of a and b. iii. Find the exact value of Solution i. We are given that; We are required to show that; Since provided that ; Therefore; If and are […]
Question The parametric equations of a curve are for t < 0. i. Show that in terms of t. ii. Find the exact coordinates of the only point on the curve at which the gradient is 3. Solution i. We are required to find for the parametric equations given below; […]
Question i. By differentiating , show that if θ then . ii. Hence show that Giving the values of a and b. iii. Find the exact value of Solution i. We are given that; We are required to show that; Since provided that ; Therefore; If and are […]
Question The parametric equations of a curve are for t < 0. i. Show that in terms of t. ii. Find the exact coordinates of the only point on the curve at which the gradient is 3. Solution i. We are required to find for the parametric equations given below; […]
Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1. i. Show that . ii. Use an iterative process based on the equation in part (i) to find the value of […]
Question The diagram shows the curve and its minimum point M. i. Show that the x-coordinate of M can be written in the form , where the value of a is to be stated. ii. Find the exact value of the area of the region enclosed by the curve and the […]
Question The diagram shows the curve , for . The x-coordinate of the maximum point M is denoted by . i. Find and show that satisfies the equation tan 2x = 2x + 4. ii. Show by calculation that lies between 0.6 and 0.7. iii. Use the iterative formula to find the […]
Question The parametric equations of a curve are i. Find an expression for in terms of t. i. Find the equation of the normal to the curve at the point for which t = 0. Give your answer in the form ax + by + c = 0, where a, b and […]
Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1. i. Show that . ii. Use an iterative process based on the equation in part (i) to find the value of […]
Question The diagram shows the curve and its minimum point M. i. Show that the x-coordinate of M can be written in the form , where the value of a is to be stated. ii. Find the exact value of the area of the region enclosed by the curve and the […]
Question The diagram shows the curve with equation . The minimum point on the curve has coordinates and the x-coordinate of the maximum point is , here and are constants. i. State the value of . ii. Find the value of . iii. Find the area of the shaded region. iv. The gradient, , of the curve has […]
Question A curve is such that i. Find ii. Verify that the curve has a stationary point when and determine its nature. iii. It is now given that the stationary point on the curve has coordinates (−1, 5). Find the equation of the curve. Solution i. Second derivative is the derivative of the derivative. If we have […]
Question It is given that , for . Show that is a decreasing function. Solution To test whether a function is increasing or decreasing at a particular point , we take derivative of a function at that point. If , the function is increasing. If , the function is decreasing. If , the test is inconclusive. We are given that; […]
Question A curve is defined for x > 0 and is such that . The point P(4, 8) lies on the curve. i. Find the equation of the curve. ii. Show that the gradient of the curve has a minimum value when x = 2 and state this minimum value. Solution i. We are given that; Therefore, for the given case; […]
Question The diagram shows part of the curve , crossing the y-axis at the point B(0, 3). The point A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C. i. Find the equation of the tangent to the curve at A. ii. Determine, showing all necessary working, whether C […]
Question The diagram shows a plan for a rectangular park ABCD, in which AB = 40m and AD = 60m. Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround a triangular playground. The length of DY is m and the length of XC is m. i. Show that […]
Question A curve has equation . Verify that the curve has a stationary point at and determine its nature. Solution i. A stationary point on the curve is the point where gradient of the curve is equal to zero; Therefore first we need gradient of the given curve. Gradient (slope) of the curve is the derivative of […]
Question An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing […]