Calculus: Derivatives Flashcards
instantaneous rates of change, derivatives, tangent lines, normal lines
What is an instantaneous rate of change?
The rate of change at a specific point on a function
What is a derivative?
The derivative of a function at a given point represents the instantaneous rate of change of the function at that point. In other words, it tells you how fast the function is changing at that specific moment.
What is the relationship between a tangent line and a derivative?
The gradient (a.k.a. slope) of a tangent line to a specific point is equal to the derivative at that point.
What is a tangent line?
A line which touches (but does not cross through) a curve at a particular point. The gradient of the tangent line is equal to the derivative of the curve at that point.
Where is the derivative of a curve equal to zero?
At the local max, local min, or inflection point
How to find the derivative of a function?
Use the power rule
What is the power rule?
1) Multiply the coefficient by the exponent
2) Subtract 1 from the exponent
What is the derivative of a constant term?
0 (when there is a constant term, it just disappears)
What is the derivative of function f(x)?
f’(x)=15x^4
Explanation:
Find the derivative
What does “differentiate” mean?
Find the derivative
What notation do we use to represent the derivative?
f’(x) “f prime of x”
dy/dx “dee why dee ex”
Differentiate the function
Differentiate the equation
Differentiate y with respect to x
How do you find the equation of a tangent line at a particular point?
1) Find the derivative using the power rule.
2) Plug in the x-value to the derivative equation to find the gradient of the tangent line.
3) Use point-gradient form of a line to find the equation.
What is a normal line?
a line perpendicular to the tangent line
What does this mean?
It means derivative. It’s just another way of writing
f’(x).
What should you do if you are asked to
“find the gradient of the graph of f at x=3 “?
- By hand (and at least written on exam for method marks):
- Take the derivative of f.
- Plug 3 into this function. Make sure to write down “ f’(3) “.
- On the calculator (to double check)
How can you use the derivative to find out on which intervals the original function is increasing/decreasing?
- When the derivative is negative (f’(x)<0), the function is decreasing.
- When the derivative is positive (f’(x)>0), the function is increasing.
What should you do when asked to “minimize”, “maximize”, or “optimize” a function?
- Option 1:
Graph the function on your GDC and use it to find the min/max - Option 2:
- Find the derivative of the function, f’(x).
- Set the derivative to zero, f’(x)=0.
- Solve for x.
- Plug in x from the previous step to find the y-value.
How are derivatives related to the max/min of a function?
The tangent at the min or max of a function is a flat line, which means its slope is 0. Therefore, the min/max occurs at f’(x)=0.
You find the derivative and then solve for x when f’(x)=0. (This gives you the x-coordinate. To find the y-coordinate, you have to plug this x into the original function.)
