Use the sliders to create the function to y = [math]2^x[/math][br]Move point B closer to point A, from both sides of point A.[br]Note the "limiting value" that the gradient of AB approaches.[br]
[size=150]Gradient of tangent to y = [math]2^x[/math] at the point (0,1)[/size][br]The limiting value of the gradient of the chord represents the gradient of the tangent to the curve.[br]What is value of the gradient of the tangent to y = [math]2^x[/math] at (0,1), correct to 3 decimal places?
Use the slider to change the function to y = [math]3^x[/math][br]Move point B closer to point A, from both sides of point A.[br]Note the "limiting value" that the gradient of AB approaches.
[size=150]Gradient of tangent to y = [math]3^x[/math] at the point (0,1)[/size][br]The limiting value of the gradient of the chord represents the gradient of the tangent to the curve.[br]What is value of the gradient of the tangent to y = [math]3^x[/math] at (0,1), correct to 3 decimal places?
It seems logical that somewhere between y = 2^x and y = 3^x, the gradient at A(0,1) will be exactly 1.[br]Use the slider to change the function y = [math]a^x[/math], and experiment by moving point B closer to point A, from both sides of point A, until you find the function that has a gradient of 1 at (0,1).[br]
What is value of "a" for y = [math]a^x[/math], to have a gradient of 1 at (0,1), correct to 3 decimal places?
Now try moving point A to different locations on this special exponential function.[br]as well as point A(0,1) will be exactly 1.[br][br]Move point B closer to point A, from both sides of point A, and note the gradient of the tangent at various positions of point A.[br][br]What does this tell you about the gradient of the tangent anywhere on the function y = e^x[br]