Monday was a full B-day schedule. Warm up with 2 graphs, had to identify which was the original function and which was the derivative. Debriefed the graphing derivative handout finished over the weekend. Looked at how to use a TI-84 to find the numerical derivative at a point (demonstration here). Used this capability to find and plot the slope values of y=sin(x) and discovered that the derivative of sin(x) is cos(x). A similar argument shows that the derivative of cos(x) is -sin(x). Added these to our booklets and did a quick example.
Then opened a discussion on differentiability, which essentially describes a function that is continuous and has smooth transitions in its slopes: no jagged corners, cusps, or vertical tangents. Introduced analogy of "mr pickle car" and his struggle to drive over nondifferentiable curves. Will explore this in more depth Wednesday.
Notes from board
Homework
p. 114 #54-69 (multiples of 3), 70-77
Resources
writing the equation of a tangent line: link
finding horizontal tangent lines (remember that horizontal lines have slope equal to zero and derivatives are slopes, so just set derivative equal to zero and solve) link
finding a value (#63 and 66) to make a line and curve tangent (made by me :)) link
Continued differentiability discussion during DS. This involved seeing graphically that continuity was a necessary but not sufficient condition for differentiability. Also looked at showing if a function was non-differentiable algebraically by seeing if its derivative function was continuous or not. Categorized types of non-differentiable points as cusps, corners, and vertical tangents. Saw that a differentiable function by rule had to also be continuous, which means that differentiability implies continuity.
Warm up in class dealt with doing the power rule in reverse (a process called integration) and recognizing the limit definition of derivative in the context of a trig derivative (-cos(x) in this case). Went over homework, then continued discussion on differentiability and how to think about it verbally, graphically, and algebraically. Worked through an algebraic example that was not continuous and therefore could not possibly be differentiable.
Introduced the product and quotient rules which are unfortunately not as straight forward as simple sums and differences. Wrote these 2 new rules into our booklets and worked through some examples of each.
Notes from board
Mr Frumble Video
Homework:
p. 105 #75-80 (determine differentiability graphically), 87-90 (determine algebraically)
p. 125 #3-33 (multiples of 3)
Resources
forthcoming