Question a. The functions and are defined for by , where and are positive constants , Given that and , (i) calculate the values of and , (ii) obtain an expression for and state the domain of . b. A point P travels along the curve in such a way that the x-coordinate of P at time t minutes is […]
Question A curve has a stationary point at and is such that . i. State, with a reason, whether this stationary point is a maximum or a minimum. ii. Find and . Solution i. Once we have the coordinates of the stationary point of a curve, we can determine its nature, whether minimum or maximum, by finding […]
Question i. Express in the form . ii. Determine whether is an increasing function, a decreasing function or neither. Solution i. We have the expression; We use method of “completing square” to obtain the desired form. We complete the square for the terms which involve . We have the algebraic formula; For the given case we can compare the […]
Question A curve is such that . The curve has a stationary point at where . i. State, with a reason, the nature of this stationary point. ii. Find an expression for . iii. Given that the curve passes through the point , find the coordinates of the […]
Question The equation of a curve is , where and are constants. i. In the case where the curve has no stationary point, show that . ii. In the case where and , find the set of values of for which is a decreasing function of . Solution i. We are given; Gradient […]
Question A curve has equation . i. Find . A point moves along this curve. As the point passes through A, the x-coordinate is increasing at a rate of 0.15 units per second and the y-coordinate is increasing at a rate of 0.4 units per second. ii. Find […]
Question The diagram shows parts of the curves and intersecting at points and . The angle between the tangents to the two curves at is . i. Find , giving your answer in degrees correct to 3 significant figures. ii. Find by integration the area of the shaded region. Solution i. Angle between two curves is the angle […]
Question The function is defined for and is such that . The curve passes through the point . i. Find the equation of the normal to the curve at P. ii. Find the equation of the curve. iii. Find the x-coordinate of the stationary point and state with a reason whether this point is a maximum or a […]
Question The base of a cuboid has sides of length cm and cm. The volume of the cuboid is 288 cm3. i. Show that the total surface area of the cuboid, A cm2, is given by ii. Given that can vary, find the stationary value of A and determine its nature. Solution From the given information we can compile following data; […]
Question A curve is such that , where a is constant. The point lies on the curve and the normal to the curve at is . i. Show that . ii. Find the equation of the curve. Solution i. If two lines (or one line and a curve) are perpendicular (normal) to each other, then product of […]
Question A functions is such that for . i. Find an expression for and use your result to explain why has an inverse. ii. Find an expression for , and state the domain and range of . Solution i. We have; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with […]
Question The diagram shows part of the curve and the tangent to the curve at . i. Find expressions for and . ii. Find the equation of the tangent to the curve at P in the form . iii. Find, showing all necessary working, the area of the shaded region. Solution i. First we find the expression […]
Question The equation of a curve is such that . Given that the curve has a minimum point at , find the coordinates of the maximum point. Solution To find the coordinates of a stationary point (in this case a maximum point) we need derivative of equation of the curve. We are given the second derivative of the equation of […]
Question A curve is such that . The curve passes through the point . i. Find the equation of the curve. ii. Find . iii. Find the coordinates of the stationary point and determine its nature. Solution i. We are given that curve passes through the point and we are required to find the equation of the curve. We […]