Determine the value of the constant a Hence, solve the equation + = 0 x3 ax –12 A function is defined by: = 3x2 + + 1. f(x) x3 x – Find and and hence state a factor of f(1) f(–2) f(x). Express f(x) in the form (x +p)(x2 + +r), where and are numbers to qx p, q r be determined. Hence solve the equation f(x) = 0 ** Show that (x is a factor of 9x2 + 26x 24. Find the set of values for x3 –2) – – which 9x2 + 26x 24 < 0. x3 – – that = 6x3 + + + 4. k(x) ax2 bx You are also told that = and = 49. k(–1) k(3) –15 Show that = and = 0 a b –13 Write, as the product of linear factors. k(x) Hence solve for if = 0 x, k(x) Given = + + 6 is exactly divisible by + Determine the values p(x) x3 ax2 bx x2 x – –6. of and a b. DIFFERENTIAL CALCULUS Outcomes for this session: By the end of this session, you should be able to: Develop an intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point. Determine the average gradient of a curve between two points, i.e.: = m Determine the gradient of a tangent to a graph, which is also the gradient of the graph at that point. Introduce the limit principle by shifting the secant until it becomes a tangent. Use first principles for (x) = for = ; and f(x) k f(x) = ax f(x) = ax + b. Use the rule for n Find equations of tangents to graphs of functions. Sketch graphs of cubic polynomial functions using differentiation to determine the ordinate of stationary points. Also, determine the of the graph using the x-intercepts factor theorem and other techniques. practical problems concerning optimization and rates of change, including calculus of motion.
Determine the value of the constant a Hence, solve the equation x3 + ax –12 = 0 A function is defined by: f(x) = x3 – 3x2 + x + 1. Find f(1) and f(–2) and hence state a factor of f(x). Express f(x) in the form (x +p)(x2 + qx +r), where p, q and r are numbers to be determined. Hence solve the equation f(x) = 0 ** Show that (x –2) is a factor of x3 – 9x2 + 26x – 24. Find the set of values for which x3 – 9x2 + 26x – 24 < 0. that k(x) = 6x3 + ax2 + bx + 4. You are also told that k(–1) = –15 and k(3) = 49. Show that a = –13 and b = 0 Write, k(x) as the product of linear factors. Hence solve for x, if k(x) = 0 Given p(x) = x3 + ax2 + bx – 6 is exactly divisible by x2 + x –6. Determine the values of a and b. DIFFERENTIAL CALCULUS Outcomes for this session: By the end of this session, you should be able to: Develop an intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point. Determine the average gradient of a curve between two points, i.e.: m = Determine the gradient of a tangent to a graph, which is also the gradient of the graph at that point. Introduce the limit principle by shifting the secant until it becomes a tangent. Use first principles for (x) = for f(x) = k ; f(x) = ax and f(x) = ax + b. Use the rule for n Find equations of tangents to graphs of functions. Sketch graphs of cubic polynomial functions using differentiation to determine the of stationary points. Also, determine the x-intercepts of the graph using the factor theorem and other techniques. practical problems concerning optimization and rates of change, including calculus of motion. ordinate