A Thinking Question About Slope Over Time (2.6 # 34)

 A Thinking Question About Slope Over Time (2.6 # 34)

Here we reflect on the how a positive derivative means something is increasing, and if it is negative, something is decreasing. We can then see an overall trend that is negative is indicating an inverse relationship between the variables (one goes up and the other goes down), and a positive trend in slope has a direct relationship (they go up and down together).  Looking at this should also get your brain in thinking about parametric relationships where both the x and y are are functions of time.  

Related Rates: Filling a Pool (2.6 # 19)

 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)

 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)

 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)

 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)

 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)

 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!

Basic Related Rates Example Quiz Solutions

 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

Implicit Differentiation (2.5 # 35, 37, 39)

 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)

 More Chain Rule Practice (2.4 #63)

The Chain rule to rescue, once again!