The second derivative
Higher Derivatives:
Functions can be derived multiple times:
f(x) derived is f'(x)
f'(x) derived is f''(x)
f''(x) derived is f'''(x)
etc...
However for the scope of high school we will only be looking at f(x), f'(x) and f''(x) or y, y', y''.
Functions can be derived multiple times:
f(x) derived is f'(x)
f'(x) derived is f''(x)
f''(x) derived is f'''(x)
etc...
However for the scope of high school we will only be looking at f(x), f'(x) and f''(x) or y, y', y''.
Now your turn!!!
The sign of the second derivative
The second derivative (y" and f"(x)) relates to how the gradient function is changing (y' or f'(x)) therefore:
if f"(x) > 0 then f'(x) is increasing
if f"(x) < 0 then f'(x) is decreasing
if f"(x) = 0 then f '(x) is stationary
NOTE: The relationship between f"(x) and f'(x) is the same relationship as f'(x) and f(x).
if f"(x) > 0 then f'(x) is increasing
if f"(x) < 0 then f'(x) is decreasing
if f"(x) = 0 then f '(x) is stationary
NOTE: The relationship between f"(x) and f'(x) is the same relationship as f'(x) and f(x).
In summary if:
f"(x) > 0 the curve is concave upwards.
f"(x) < 0 the curve is concave downwards
f"(x) = 0 and their is a change in concavity it is a point of inflexion.
How do you think you'd check that concavity changed?
Complete Exercises below:
f"(x) > 0 the curve is concave upwards.
f"(x) < 0 the curve is concave downwards
f"(x) = 0 and their is a change in concavity it is a point of inflexion.
How do you think you'd check that concavity changed?
Complete Exercises below: