Mathorama: Chapter 2

Showing posts with label Chapter 2. Show all posts

Sunday, October 17, 2021

Monday, October 5, 2020

Related Rates: Filling a Pool (2.6 # 19)

I encourage beginners to organize related rates word problems with these 5 steps:

1: Find what?

2: Given what?

3: What is the connection between these?

4: Implicitly differentiate something.

5: Plug in what you know, solve for what you what to find (remember the units!)

Related Rates with The Volume of a Cone (2.6 #17)

A related rates word problem has a lot of pieces, so try organizing your strategy:

1) Find what?

2) Given what?

3) Connect the given with what you need to find with an equation

4) Differentiate something

5) Substitute what you know, solve for what you what to find.

Related Rates (2.5 #5)

We differentiate both sides of the equation with respect to time. (After all, a "rate" is something over time)

TI-84 Tips to Show Curves are Orthogonal To Each Other (2.5 #63)

Here we use a TI-84 to show how two curves are orthogonal to each other. If they are, then the tangent lines's slope at the points of intersection are perpendicular, and be negative reciprocals of each other:

If the slope is a/b, then the perpendicular slope is -b/a. The calculator has a (CALC)5. intersect feature and a (CALC)6. dy/dx feature, so this is how we can tell if the curves are orthogonal.

Tangents to a Circle (2.5 #57)

While this is easily done with basic algebra, let's confirm our result with implicit differentiation!

A Second Derivative using Implicit Differentiation (2.5 #49)

The second derivative is just the derivative of the first derivative. When we differentiate implicitly two times, we can sometimes take advantage of substitution of a previous result to simplify our result with ease!

Sunday, October 4, 2020

Basic Related Rates Example Quiz Solutions

These examples use the formulas of the area and alums of circles, spheres, and cones. These techniques are applicable to cubes, cylinders, rectangles, triangles, prisms and pyramids using their respective formulas. I encourage beginners to organize the given information by these 5 steps:

1: Find what?

2: Given what?

3:What is the connection between these?

4:Implicity differentiate something

5:Plug in what you know, solve for what you don't know

Here is a pdf of these 5 problems and their solutions

Thursday, October 1, 2020

Implicit Differentiation (2.5 # 35, 37, 39)

We don't need to "solve for y" before we take a derivative. This allows us to handle relationships that are not functions like these three examples. The only thing to be mindful of it to use the chain rule!

Do a derivative of y^2 is not merely 2y, it it 2y*(dy/dx) or if you like 2y*y'.

Finding Derivatives Based on a few facts (2.4 #98)

Finding Derivatives Based on a few facts

Using a table of values to find a derivatives at a particular value of x.

More Chain Rule Practice (2.4 #63)

The Chain rule to rescue, once again!

Tuesday, September 29, 2020

The Chain Rule in Action and Confirmed with our TI-84

We attack a sec^3 function with the Chain Rule and confirm our slope with a TI-84

We attack a sec^3 function with the Chain Rule and confirm our slope with a TI-84. The show the derivative of sec x is tan x * sec x. Next we use the chain rule to twice to find the slope at x=0.

Tangent Line to A Circle with the Chain Rule (2.4 #79)

He we find the slope of a semi-circle using the chain rule, and check our tangent line equation with a TI-84 style calculator.

Sunday, September 27, 2020

Higher Order Derivatives (2.3 #93, 103, 115)

The rate of change of the rate of change is the derivative of the derivative, aka the "Second derivative". We we write it " f''(x)" or sometimes "d^2x/dy^2". But there is reason to stop there. The derivative of the second derivative is the third derivative, and so on. We will even find an eighth derivative in this video!

Rate of change of Area per second (2.3 #83)

Here we look at the rate of change of area per second which requires us to make a function (the Area of a rectangle) and compute the rate of change of that function. This gives us a glimpse of what Calculus "word problems" look like!

Can Different Functions Have the Same Derivative? (2.3 # 79)

Can Different Functions Have the Same Derivative? (2.3 # 79)

Here we investigate a claim that two function have the same derivative. We use the quotient rule to confirm it, then looking at the graph, we see why these functions might actually have the same slope.

Quotient Rule to Find a Tangent Line with a Certain Slope (2.3 #77)

Quotient Rule to Find a Tangent Line with a Certain Slope (2.3 #77)

After using the quotient rule to find a formula for slope, we set it to a certain slope to find when the function has that slope. To get the equation of the line, we use the original function to find the y coordinate, and place it in the "Point-Slope" form of a linear equation: y-Y = m (x-X) where the point is (X,Y) and m is the slope.

Finally we use the calculator to check our work both graphically as well as numerically.

Check your Derivative at a Point with a Calculator (2.3 # 59)

Check your Derivative at a Point with a Calculator (2.3 # 59)

After finding a derivative at a point with the quotient rule, we check our work with a TI-84 style calculator.

Thinking Questions with Derivatives (2.2 #67,79)

Thinking Questions with Derivatives (2.2 #67,79)

With the Power (or exponent) Rule, it's easy to handle some questions that have to do with slope! We also use the TI-84 to confirm our result.

More Quotient and Product Rule (2.3 #35, 51, 55)

More Quotient and Product Rule (2.3 #35, 51, 55)

Here the calculus is easy, but sometimes you have to use algebra to match your answer with the answer in the back of the book. A great skill to have so you can select a multiple choice item that expresses the answer in a slightly different form than the way you have it....

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