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Resit CA

Task 1: Meaning of a Derivative

Follow the instructions below to create an object illustrating the geometric meaning of a derivative. 1. Show the Axes and change the x-axis to y-axis ratio to 1:4 2. Create the function f(x) = x^2 3. Create a slider called h that runs from 0.01 to 4 and set it to 4 4. Create a point A, on the function f 5. Create the point B=(x(A)+0.4*h,f(x(A)+0.4*h)) 6. Create the point C=(x(A)+h,f(x(A)+h)) 7. Draw a line through A and C 8. Find the tangent tool under the Construct heading, click on the point B and then the function f 9. Find the slope tool under the Measure heading and click on the line through C. This slope should be called m 10. Find the slope of the tangent at B, this slope should be called m1 11. Hide the slopes m and m1 12. Create a textbox that says 'Slope of AC=m' and another that says 'Slope of tangent = m1' Note that m and m1 should be the variables in orange boxes. 13. Create a textbox, click the LaTeX formula button and enter: \text{Derivative} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \text{Slope of AC} Move h towards 0 and examine what happens. Change the colours of your lines to make the construction clear.

Task 2: Asymptotes

Create sliders a and b that range from 1 to 5. Create the function f(x)= ax/bx+1 and put the function inside a checkbox called "Original Function". Plot the vertical & horizontal asymptotes of this function and label them as "Vertical/Horizontal Asymptote". Find the point of intersection of these two asymptotes and label it as "Centre of Symmetry". Reflect the function f through this point and put the resulting function inside a checkbox called "Image."