## Calculus 12 Derivatives

**Specific Curriculum Outcomes**

**A2**Demonstrate an understanding of the definition of the derivative.

**C4**Demonstrate an understanding of the connection between the graphs of

*f*and

*f*’.

**B5**Find where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents.

**B6**Derive, apply, and explain power, sum, difference, product and quotient rules.

**B7**Apply the chain rule to composite functions

**B8**Use derivatives to analysis and solve problems involving rates of change.

**B9**Apply the rules for differentiating the six trigonometric functions

**A3**Demonstrate understanding of implicit differentiation and identify situations that require implicit differentiation

**B10**Apply the rules for differentiating the six inverse trigonometry functions

**(optional)**

**B11**Calculate and apply derivatives of exponential and logarithmic functions

**B12**Apply Newton’s method to approximate zeros of a function

**(optional)**

**B13**Estimate the change in a function using differentials and apply them to real world situations

**B14**Solve and interpret related rate problems

## activities

**Introducing Calculus Activities from Underground Mathematics**- A number of rich tasks for introducing calculus concepts design at the University of Cambridge.

**Tutorials for the Calculus Phobe**- A great series of videos on limits and continuity that very clearly describe these ideas in an approachable and interesting manner.

**Derivative Matching Cards from David Petro**- Students are give a set of cards with the graph of either a linear, quadratic or cubic function on them. Their job is to pair up the graphs of the functions with the graphs of the derivatives. There are a total of 12 functions with 12 derivatives. There is a Desmos Card Sort Activity that was inspired by this activity.

**Related Rates, Yet Another Redux from Sam Shah**- Sam has some great ideas on introducing related rates. He has a very nice worksheet (Dan Meyer even commented on it). I really like his idea of using balloons. He started class asking for a volunteer to blow up balloons. He taped an empty balloon, a balloon with one breath, with two breaths, etc to the whiteboard. Then the class had a discussion about what they could measure about the balloons and what changed with each balloon. This five minute start to class reinforced the main idea (hopefully). We changed one thing. It changed a bunch of other things. But just because one thing changed in one particular way doesn’t mean that everything changed in that same way. For example, just because the volume increased at a constant rate doesn’t mean the radius changed at a constant rate.

**Which One Doesnt Belong (WODB) Problems****-**You can use the problems to discuss continuity and differentiability.

**A fun way to Estimate the value of e from Mike Lawlor**- Generate 64 random integers from 1 to 64 and put snap cubes on the squares of the chessboard to represent those numbers. How many of the squares end up with no cube on them? The ratio of 64 to empty squares is approximately e.