advertisement

MEI Conference 2015 Active learning with GeoGebra Tom Button [email protected] www.mei.org.uk/geogebra Differentiation 1 – Exploring the gradient on a curve 1. In the input bar enter a cubic function: e.g. f(x)=x^3-2x^2-2x+2 2. Use New Point (2nd menu) to add a point on the curve. 3. Use Tangent (4th menu) to create a tangent to the curve at point A. 4. Use Slope (8th menu) to measure the gradient of the tangent. 5. Plot the gradient function by entering g(x)=f '(x) in the input bar. You might find it easier to see if you change the gradient function to a red dotted line using the Graphics Styling bar. Question for discussion How is the gradient of the tangent (as the point moves) related to the shape of the gradient graph? Verify your comments by trying some other functions for f(x). Problem Change your function in GeoGebra so that is has the following gradient functions: Extension Task Find the point on the function f( x) x3 6 x 2 9 x 1 where the tangent has its maximum downwards slope. Investigate the point with maximum downward slope for other cubic functions. 1 www.mei.org.uk/geogebra Differentiation 2 – Stationary points 1. In the input bar enter a cubic function: e.g. f(x)=x^3-2x^2-2x+1 2. Find the turning points of the function: in the input bar enter TurningPoint[f] 3. Plot the gradient function by entering g(x)=f '(x) in the input bar. You might find it easier to see if you change the gradient function to a red dotted line using the Graphics Styling bar. Question for discussion How can you use the graph of the gradient function to explain why the function has a local maximum at A and a local minimum at B? Verify your comments by trying some other functions for f(x). Problem (Try this on paper first then check your answer on GeoGebra) Find the values of x for which the following functions have turning points and determine whether they are maxima or minima: f( x) x3 3 x 2 9 x 3 g( x) x 4 4 x3 36 x 2 8 Extension Task Use GeoGebra to find the gradient function of f( x) x3 6 x 2 12 x 5 . Explain why the function has a stationary point that is neither a maximum nor a minimum (a stationary point of inflection). Find some other functions that have stationary points of inflection. 2 www.mei.org.uk/geogebra Integration: Area under a curve It is essential that this is entered as a function f(x). 1. In the Input bar enter: f(x)=x^2 2. Use Slider (11th menu) to create a slider for a. 3. In the Input bar enter: A=Integral[f, 0, a] Create the slider with minimum value 0. Question for discussion What is the relationship between the area and the value of a? What is the relationship if f(x) is changed to a different power of x? Problem (Try this on paper first then check your answer on GeoGebra) Find the area under f(x) = x5 between x = 0 and x = 3. Further Tasks Add a Slider (11th menu) for b. Investigate the area under f(x)= xn between x = a and x = b. Investigate the areas under functions that are the sums of powers of x: e.g. f(x)=x³ + 3x² + 4x +1 3 www.mei.org.uk/geogebra Functions: Transformations 1. Use Slider (11th menu) to create sliders for a and b. It is essential that this is entered as a function f(x). 2. In the Input bar enter: f(x)=x^2 3. In the Input bar enter: g(x)=f(x+a)+b Question for discussion What transformation maps f(x) onto g(x)? Does this work if other functions are entered for f(x)? Problem (Try this on paper first then check your answer on GeoGebra) Show that f(x) = x4 – 8x³ + 24x² – 32x +13 can be written in the form (x+a)4 + b and hence find the coordinates of the minimum point on the graph of y = f(x). Further Tasks Use Slider (11th menu) to create sliders for c and d. In the Input bar enter: h(x)=c*f(d*x). What transformation maps f(x) onto h(x)? Investigate g(x) and h(x) for f(x)=log10x. NB this is entered as: f(x)=log10(x) 4 Changing f(x) to f(x)=x³–x might help make it clearer. www.mei.org.uk/geogebra AS Mathematics Construction Problems Example 0. Create two points A and B on the x axis. Construct a quadratic graph that passes through A and B. A possible solution (there are other ways to do this) Use the New Point button to add points A and B fixed to the x-axis. In the Input bar define two variables: a = x(A) and b = x(B). Define a new curve on the Input bar: y = (x–a)(x–b) Problems 1. 2. Create points A, B and C fixed to the x-axis and D fixed to the yaxis. Construct a cubic that passes through A, B, C and D. Create a triangle with one point on the origin and one point on the x-axis. Construct circles centred on each vertex such that all three circles touch each other. 3. 4. Plot the curve f(x)=x² and add a point A to the curve. Construct the point B such that the tangents at A and B are perpendicular. Create two points A and B. Construct a cubic that has stationary points at A and B. 5 www.mei.org.uk/geogebra