Supplemental Topics, Branching out more with Calculus l

Chapter 8: Supplemental Topics:

 

Identifying the graphs of f(x), f’(x) and f’’(x).

When examining the graph of f(x) to find f’(x) and examining the graph of f’(x) to find f’’(x):

• Does the function’s curve move downwards (decreasing) going right from that point to its rightmost point (in which case the slope (y-value of the derivative) would be negative) or does the curve go upwards (increasing) from that point to the rightmost point (in which case the slope (y-value of the derivative) would be positive). 

• The steeper the slope is, the greater the absolute value of the y-value of the derivative function will be. 

Limits (L’Hopital’s Rule) and Related Rates:

When we are relating the rate of two functions f(x) and g(x), we consider the following:

For indeterminate forms, recall L’Hopital’s Rule. 

L'Hopital's Rule: derive the numerator and denominator (keep it in fractional form), then plug in the value the limit approaches. If after plugging in the value, you come back to the original limit, try the rule again. If it happens to be in the exact same form again, that means that there may be no proper conclusion. 

Reaching back to F.T.O.C. and exploring the graphs of F(x), and F’(x).

Since F is just a function that refers to the area under the curve f(x), we look at the behavior of the area under the curve when we are examining F. 

Area above the x-axis within an interval is considered a positive area. F(x) is a y-value = the area from a starting point of “a” to “x”. If the area remains positive from “a” to “x”, and “x” is the x-value at which f(x) = 0, (assuming the function’s next rightmost point from “x” would be a negative y-value), then we know the area is getting bigger in value from “a” towards “x” and getting more positive and when it gets to “x”, it is at its max. 

Area below the x-axis within an interval is considered a negative area. If the area remains negative from “a” to “x”, and “x” is the x-value at which f(x) = 0, (assuming the function’s next rightmost point from “x” would be a positive y-value), then we know the area is getting smaller in value from “a” towards “x” and getting more negative and when it gets to x, it is at its min. 

Taking the derivative of F, or F’(x) we get our original curve f(x):

Discrete data: just data points, or points that don’t necessarily follow a function/ graph. 

Put them on a graph to help with visualization! This will also help with estimating different types of area. Make sure you know what the area really implies, so that you can put correct units! Find what f(t)’s units are and dt’s units are and multiply them to find the units of the area. 

Recall the ways to approximate area:

• Left/Right areas 

• Trapezoid method

• Simpson’s method

*** To find minimum and maximum units of area, use the left/right sums (identify which gives underestimates and which overestimates).***

Averages:

Velocity from any (d1, t1) to (d2, t2) = change in distance (d2 - d1) / change in time (t2 - t1)

Average velocity = change in total distance (d(t) - d(0)) / change in total time (t - 0), where “t” represents the last time value. 

Acceleration from any (v1, t1) to (v2, t2) = change in velocity (v2 - v1) / change in time (t2 - t1)

Average acceleration = change in total velocity (v(t) - v(0)) / change in total time (t - 0), where “t” represents the last time value. 

Average of value of a function:

Recall…

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