Calculus Interpretation: Part 1

Transcription

1 Saturday X-tra X-Sheet: 8 Calculus Interpretation: Part Key Concepts In this session we will focus on summarising what you need to know about: Tangents to a curve. Remainder and factor theorem. Sketching a cubic function. Graph interpretation. Terminology & definitions Tangent: A line drawn to a curve with only one point of contact. The equation of a tangent to a curve To find the equation of a tangent to a curve We must have the coordinate of the point of contact. If not, we must have the value and substitute back for the y; OR; we must have the gradient and work back to find the value. To find the gradient, we find the derivative and substitute the value of the point of contact. We can now find the equation of the tangent using y = m + c or y y = m ) X-ample Questions ( Tangent to a curve. Calculate the equation of the tangent to the curve = at =.. Determine the value of k for which the equation of the tangent to = + k + 4 at the point ( ; y ) is y = 8 8. Remainder and factor theorem Remainder Theorem If a polynomial f () ( a b) then b f = a Re mainder Factor Theorem If a polynomial f () ( a b) and b f = a 0 ( a b) f () is a factor of

2 X-ample Questions. Divide = + + by ( + ). Given g( ) = 6 a) Show that ( ) is a factor of g (). b) Factorise g () completely. c) Hence solve for if g ( ) = 0.. Given = a) Determine all values of such that f() = 0 b) Hence or otherwise, solve (-) + (-) 7(-) = 6 Sketching a Cubic Function Shape a>0 y = a + b + c + d or if a<0 Stationary points / = 0 The -intercepts = 0 The y-intercepts =0 The point of inflection // = 0

3 X-ample Questions. = is given. a) Show that ( + ) is a factor of f (), and hence, determine all the factors of f (). b) Determine the co-ordinates of the turning points of f. c) Draw a neat sketch graph of f, showing all the and y intercepts, as well as the turning points. f = +. Consider the function: ( ) a) Show that ( ) is a factor of. b) Factorise fully and hence solve for i c) Find f ' ( ) f =0. and hence the co-ordinates of any local maimum/minimum points. d) Determine the co-ordinates of the point of inflection. e) Sketch the graph of on the answer sheet provided showing all turning points and intercepts with the aes. f) Determine the equation of the tangent to the curve o Finding the equation of a cubic function If we care given the intercepts and one other point we use y = a )( )( ) ( X-ample Questions. Find the equation of y f where =. -. Find the equation of y 4 -

4 X-ercise. a) Solve for b) Hence solve for. Below are the graphs of = and g(), a cubic function. The two functions have roots at A and B, and g() has another root at =. The length of DE = 6 units. a) Find the roots at A and B. b) Give the coordinates of E. c) Find the equation of the function g(). d) Find the coordinates of K, where the two functions intersect. e) Does F, the turning point of g(), lie on the ais of symmetry of f()? Show all working. f) There are two values where the two functions are increasing at the same rate. Find these values in surd form.. Given = + + calculate the equation of the tangent to the curve f where the gradient equals -. g( ) = 4. Given: a) Find the derivative, g ( ) from first principles. b) Hence find the equation of the tangent to g() at the point where =. 5. g ( ) = 5 a) Find the value of g (-) b) Solve g() = 0 c ) Hence, draw the graph of g, showing the coordinates of turning points, and the intercepts with the aes.

5 4 y = + and y = a + b + c 6. The curves with equations have the following properties: 7. a) there is a common point where = b) there is a common tangent where = c) both curves pass through the point (;6). Find the values of a, b and c. a) Calculate the values of a, b and c if = + a + b + c is the equation of the given graph. b) Determine the average gradient of the line through the points where = and = if it is given that a =, b = 5 and c = 6. c) Determine the coordinates of the non-stationary point of inflection. d) Write down a question for which " f ( ) < 0" could be used to determine the solution. f ( ) = Given a) Find the value of f() b) Hence, or otherwise, find the value of the -intercepts of the graph of f() c) Determine f ' () d) Find the coordinates of the turning points of the graph of f(). Clearly indicate which point is a maimum and which point a minimum e) Sketch the graph of f(), showing clearly the turning points and the intercepts with the aes f) Determine -value of the point(s) of inflection of the graph of f()

6 Answers a) ( m )(8m )( m + ) b) =0 or =- a) A (-;0) or B(-;0) b) E(0;-) c) y = d) = e) No f) = ± ) y = + 4 4a) g ( ) = / 4b) y = + 9 5a) g( ) = 0 5b) =- or = 6) a= b=-5 c=0 7a) = b) -4 7c) 68 ; 7 7d) When is f() strictly decreasing? 8a) f ( ) = 0 8b) = 0r =- 8c) / = 8 8d) 500 A ; 7 ma and B (;0) min

7 8f) A y 8 - B 8f) = 4