Differentiation Flashcards
What is the gradient of the tangent equal to
The gradient of the curve at a specified point
Find the gradient and equation of the tangent to y = 3x^2 at x = 1
y = 6x - 3
Method for finding equation of a tangent
- differentiate the equation given
- sub x into differentiated equation to find m
- find y, state x and y in the form (x, y)
- find equation of tangent by subbing known data into equation
A parabola has the equation y = x^2 - 4x + 1. Calculate the gradient and equation of the tangent at x = 3.
y = 2x - 8
The gradient of a tangent to the curve y = x^4 + 1 is 32.
a) find the point of contact of the tangent
b) find the equation of the tangent
y = 32x - 47
Find the interval for which the function y = 3x^2 + 2x - 5 is increasing
function is increasing when x > -1/3
Finding interval for when a function is increasing method
- find derivative
- set derivative > 0
- solve for x
- state interval
Finding interval when function is decreasing method
- find derivative
- set derivative < 0
- solve for x
- state interval
If a graph slopes down from left to right, the function is ________
Decreasing
If a graph slopes up from left to right, the function is ________
Increasing
The derivative of a function is equal to what
The gradient
If the derivative > 0, the curve is what
Increasing as m is positive
If the curve is increasing, what is the value of the derivative
> 0
If the derivative < 0, the curve is what
The curve is decreasing as m is negative
Value of derivative when curve is decreasing
< 0
Method for determining if function is increasing or decreasing
- find derivative
- sub x into derivative
- stare if function is increasing or decreasing
Is y = x^2 - 5x increasing or decreasing when x=4?
Increasing at x = 4 since dy/dx > 0.
(= 3)
Stationary point
Graph is neither increasing or decreasing
Gradient at stationary points
0
Nature of stationary point
- Maximum turning point
- Minimum turning point
- Point of inflection
- Maximum turning point
- Minimum turning point
- Point of inflection
Nature of turning point Minimum
Method for finding stationary point
- differentiate and set equal to 0
- solve to find x
- nature table
- sub x into original equation to find y