Calculus

In terms of growth and time

"21 mm/day is the average rate of growth between the age 2 days and age 10 days." Or "The average rate of growth of the slot between age 2 days and 10 days was 21 mm/day" "The instantaneous growth (or in general terms the slope of the tangent line at a specific point) of the Zylot at exactly 2 days was 13 mm/day

Derivative

Always spits out instantaneous rate

Slope of a secant line is to

Average rate of change of growth joining two point. Two points.

F'(x)

Can be slope of a tangent line, derivative, or instantaneous rate of change/growth.

Slope

Change in Y/Change in X

Quotient rule

D{f(X)/g(X)} = g(x)f'(x)-f(x)g'(x) _________________ {g(x)}^2 Lo Dee hi - hi Dee lo _____________________ Lo * Lo

The slope of the tangent line (general f(x))

For numbers only, no units the slope of the tangent line is given at ONE point. This is also the instantaneous rate of change. The derivative will also give us the slope of a tangent line. We can write this as: "The slope of the tangent line is 13 at point (2,f(2))" Or "13 is the slope of the tangent line to the graph at (2,f(2))" This is the instantaneous rate of change.

Slope of a tangent line is to

Instantaneous rate of change or growth if given a specific unit at and exact point or time. One point

The average rate of change equals simple example

One one test you score an 80, another you score a 90. The AVERAGE of these scores are 85. This could be the average rate of change when we take 90-80/2-1 this = an average 10 points per test (units are very important. Top is points and test is on bottom)

If asking for parallel slope of a line

Set derivative to the same slope m = 3 turns into f'(x) = 2x + 1. Parallel slope now gives you 3 = 2x + 1. Solve for x. We now have input to place into the original function.

The average rate of change equals (general f(x))

The slope of the SECANT line (This is the line between two points or joining two points). !Must have two points. Not one for average! We can write this as: "The slope of the secant line joining (2, f(2)) to (10, f(10)) is 21 units (there are no units given in the general form) OR "21 is the slope of the secant line that joins (2, f(2)) to (10, f(10))." This is now the average rate of change

Horizontal tangent lines

When the tangent is horizontal, it is of tremendous importance. It offers min/max or high/low this is when it is applied. Finding the horizontal tangent line means you are finding the points on the graph where the tangent line is horizontal. Method: horizontal lines have slope 0 there is 0 change in y. So we solve for the equation in f'(x) = 0. Then we plug that x into the original f(x) function to find P. Example: "find all points on the graph of f(x) where the tangent line is horizontal." f(x) = x^2 - 4x + 11 f'(x) = 2x - 4 Slope of the tangent line is 0 --> f'(x) = 0 So 0 = 2x - 4 so x = 2 P = (2,f(2)) = (2,7) We can say at P = (2,7) the slope of the tangent line is horizontal.

Equation of a tangent line

You have a point and a slope