### Learning Outcomes Assessed:

L01 : Determine the limit of functions and then proceed to discuss the continuity of functions

L02 : Use derivatives in applications such as tangent and normal lines, Newton’s method, related rates of change, L’Hopital’s rule, problems involving maximum and minimum values of a function and graph sketching

### Question 1

Newton and Leibniz calculus controversy. Here are some topics you should include:

**(a)** Give biographical information about Newton’s life such as when and where he lived, what he was like as a person, etc.

**(b)** Give biographical information about Leibniz’s life such as when and where he lived, what he was like as a person, etc.

**(c)** Discuss Newton’s contribution to calculus.

**(d)** Discuss Leibniz’s contribution to calculus.

**(e)** Discuss how their view of the calculus differs from today’s view.

**(f)** Discuss the controversy over the invention of calculus.

**(g)** What was the opinion at the time? What is the opinion today?

### Question 2

Your group has been hired to design the 300-foot straight stretch for Genting Highland’s 20th Century Fox new roller coaster. Design various alternatives and then discuss which one you think is the best. Write a summary as if you were trying to convince them to build your coaster, since your design is the best! Your roller coaster should fulfill these criteria:

**a)** The roller coaster should be a piecewise function with at least 5 pieces that is differentiable and continuous at all points. You need to show the graph and a proof that the function you generate is continuous and differentiable at the points where the curves split into each piece. You may use any type of function we have discussed including linear, trigonometry, exponential, logarithmic, inverse trigonometry and polynomial.

**b)** The design must begin with a 45o incline, and there must be a supporting beam every 10 feet

**Report criteria:**

**a)** Research (on the internet) the best roller coasters in the world, looking for answers to some of the following questions (and looking for inspiration when designing your own roller coaster): What makes them great? Where are they? What kind of names do they have? What makes them safe? How much does it cost to ride them? How much did it cost to build them? How fast do they go? As a group, if you could ride one roller coaster in the world, which one would you most like to ride and why?

**b)** Each person in your group should design his or her own roller coaster (given the constraints listed above). Choose a theme for your coaster and make a drawing of the coaster. Then answer the following questions:

**i)** Where is the path increasing and decreasing?

**ii)** Where is it concave up and down?

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