Differentiation - a starting investigation
- Mr Creamer
Every point of a curve has a tangent, and the tangent has a gradient. Recall for y = mx + c the line has a gradient of m. Lets investigate the gradient function, the new function we get from the gradient of curve f. 1. Drag the slider a to see what happens to the tangent to the curve at the point (a,f(a)) 2. Select show gradient value to see a blue point which shows (a, f'(a)) - the x value is a, the y value is the gradient (or slope) of the tangent 3. "set trace" and now the blue points remain as you move the slider a. 4. Can you see a pattern in the trace points? This line or curve shows the gradient of the tangent to the curve y=f(x) for all values of x and is written f'(x) (f prime of x, sometimes called f dash). We can call it the gradient function. 5. what is f'(x)? have a guess - say f'(x)=4x. Type your guess into the red input box and see if you were right. (you can hide/show the guess) When you have got the fright function for f'(x), make a note and move on to the next f(x) function (see below) 6. Try another function for f(x) by typing, say x^3 in the box.
This is an investigation, so look for patterns and try to predict what the next answer will be. 1. Find f'(x) when a) f(x) = b) f(x) = c) f(x) = d) guess for f(x) = (and predict for n=7) 2. Find f'(x) when a) f(x) = b) f(x) = c) f(x) = 3. Advanced Find f'(x) when a) f(x) = 1/x b) f(x) = 1/x^2 c) f(x) = sin(x) d) f(x) = cos(x) e) f(x) = f) guess for f(x) = ln(x)]