EXAM I: Introduction to the Derivative

1. Students should have problem-solving understanding of the relationship between the geometry of lines in the plane and analytic expressions (i.e. equations) describing them. In particular they should be able to calculate equations of lines from point or point-slope, and information about intersection, parallel and orthogonal relationships. Conversely they must be able to deduce the corresponding geometric information from the analytic.”
2. Students should know the quadratic formula and be able to calculate the point(a) of intersection of two quadratics, a quadratic and a line, line and circle, etc.

3. Students should be able to correctly and precisely define/describe the derivative of a function at a point in both geometric and analytic terms. They should understand that the analytic form represents the limiting value of the slopes of secant lines.
4. Students should be able to calculate the derivative of a simple function (linear , quadratic, f(x) = a/(x-b) ) at a given point as a limit (i.e. using the analytic definition)
5. Students should know the standard forms of the equation for circle with a given radius about a given point and should be able to find the tangent line to a circle a point. That is they should be able to geometrically (graphically) calculate the derivative of a function of the form f(x) = a +/- sqrt(r^2 – x^2)
6. Students should be able to estimate the slopes of tangent lines to a simple graph and use this information to make a table of estimated values of the derivative of the function.
7. Students should know the basic rules of differentiation (linearity, product rule, quotient rule, and power rule) and should be able to calculate the derivative of any reasonable polynomial with rational exponents or rational function of such expressions [for test 3 and 4 this extends to expressions involving compositions of exponentials and logs]. Test I problems will require linearity, quotient rule, power rule, product rule – they will not require the full chain rule but the chain rule can be used if students know it.
8. Students should be able to calculate the derivatives of simple, piecewise defined functions with components of the types described in the previous item. They should know that endpoints can cause problems but are not responsible for end-point behavior of derivative functions.
9. Students should be able to calculate the equation of the tangent line to the graph of any of the functions described in the previous two items.
10. Students should understand that knowing the tangent line at a point is equivalent to knowing the derivative and functional value at that point and be able exercise this understanding in problem-solving.
11. Students should be able to calculate and graph the derivative of the absolute value (or similar piecewise defined) function.
12. Students should understand that the tangent line is the “best” linear approximation to the curve. Students should understand the idea of tangent line approximation is simply to replace the function by the tangent line function and ask the question about the latter. Test problems will ask for estimates of functional values.
13. Students should be able to understand and execute informal limit calculations – they should understand that 0/0 and infinity/infinity are meaningless expressions.
    They should know that to exist limits must exist from both sides.
    They should know the concept of continuity and its proper definition: continuous functions are the ones for which you can actually do the intuitive thing. They should be able to explain in analytic and geometric terms why certain simple functions are or are not continuous at certain points
14. Students should be able to define and calculate in simple examples the average rate of change of a function over some interval in its domain.
15. Students should understand the interpretation of the derivative as rate of change - and should understand the derivative as the limiting value of the average rate of change. For linear motion they should understand the interpretation of this as the instantaneous velocity at a point and view it as the limiting value of the average velocity over time intervals beginning and ending at that point.