Test 1: Concept Review

• definition of a function; domains and ranges of functions; symmetry (even and odd functions); step functions, and functions defined piece-wise; vertical line test for a function.
• Categorizing types of functions (e.g. trigonometric functions, polynomials, rational functions, power functions); simple properties of trig functions; shapes of functions.
• tangent and secant lines; average and instantaneous velocity.
• limits; left- and right-handed limits; infinite limits and vertical asymptotes; the limit exists if and only if both left- and right-hand limits exist, and are equal; the limit can exist without the function being defined at a point.
• continuous at a; greatest integer function; standard intuitive theorems concerning limits (e.g. limit of a sum of two functions is the sum of the limits); squeeze theorem.
• function continuous on an interval; varieties of discontinuities (jump, removable, infinite); standard intuitive theorems concerning continuity (e.g. sum of two continuous functions is continuous); families of continuous functions; Intermediate Value Theorem; continuity of compositions.
• slopes of tangent lines as limits; slope of a curve at a point; position function; increments; average and instantaneous rates of change.
• derivative of a function at a point; estimating derivatives from tabular data and from graphs.
• derivative of a function as a function; operation of differentiation; differentiable on an interval; vertical tangent line; differentiability implies continuity: that is, if a function is differentiable somewhere, then it is certainly continuous there; reasons why a function fails to be differentiable at a point.

Test 1: Sample problems

1. Sketch the derivative of the function given by the following graph:

    

2. For each of the following functions:
   1. find the domain.
   2. find the regions for which it is continuous, and differentiable.
   3. indicate what type of function it is.
   4. does it possess any symmetry?

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3. Sketch the following function, and draw its derivative from the graph:

   

   Using the definition of the derivative, calculate its slope at x=1. Find the tangent line at that point, and draw it on your graph of f.

4. Using the definition of the derivative function, find the derivative of the function

   

5. Evaluate the limits, and justify each step by indicating the appropriate limit laws:

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6. GIven the following table of a growing population:

   

   Graph the data, and estimate the rate of change in size at day 2 in two different ways:

   • using the slope of a secant line, and
   • fitting a function to the data, sketching the tangent line, and estimating the slope.
   How might you estimate the slopes at days 1 or 3 using secant lines?