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Calculator Starter Exercises

Calculator Starter Exercises here andArithmetic Review Problems in Volume 2, Three Skills for Algebra, offer numerical experience and setting for the following lessons. They also provide a chance for teachers and tutors to check arithmetic skills.

That being said, the use here of calculators to show how to obtain the values of powers and roots  provides a plug and play view of mathematics. In earlier times, a plug and play approach might come from the use of tables of logs and exponential functions - previously called antilogs. The introduction of calculators and tables into higher mathematics departs in practice from the self-sufficiency ideal for mathematics.

In this introduction, we may rely on calculator use except in some numerical circumstances where the n-th root or power has an exact representation. See exercise 4 below.

1. In numerical exercises or illustrations,  first  table of values can be used to plot points on the graphs  y = x²  and y = x³  and second the rest of the graphs can be obtained by interpolation. The horizontal line method for solving equations from their graphs implies (i) that the equation  x²  = a has one or two solutions for a > 0 and none for a < 0; and (ii) that the equation  x³  = a has one solution for each real number a, and that further   if  a < 0 and X³  = |a| then  x = - X. 
   

   Teachers: Earlier exercises with graphs of  y = x²  and y = x³ over the intervals [-2,2] and [-3,3] may be useful here. If we put exponents and radicals before functions and relations, there would be need to discuss graphing and the mixed or impure mathematics assumptions about the interpolation of calculated points in order to obtain the full graphs for formulas y = x²  and y = x³, etc.
   

2. Numerical exercises or illustrations in the aforementioned Arithmetic Review Problems.  imply the n-th root of b > 0  be written in the power form  
   
   x = b^(1/n)  = exp((1/n) ln(b))
   
   is a positive solution of the equation  xⁿ  = b  Now for n real and n an odd natural number,  the n-th root of a nonzero real number b is given by 
   
   b^(1/n)  = sign(b) exp((1/n) ln(|b| ))
   
   The latter agrees with b^(1/n)  = exp((1/n) ln(b)) when b > 0, and extends the formula when b < 0.
   

3. Square Roots - How to do exact calculations with Webvideos.. Examples here show how to simplify or represent square and cube roots of whole numbers  with and without the aid of prime number factorization of the latter. The simplification here may be a cosmetic convention or fashion in mathematics that leads students and teachers to answers of the same form, the so-called simplified form.
   

4. Problem:  Show a link between the graph of y = ln(x) and y = e^x  via the following steps - an illustration of the reflection across a certain line between the graph of a function and its inverse.

   (2 points)  (i)  Use the ln(x) button on your calculator to fill in the table (or a copy of it) 

   x	1/6	0.2	0.25	1/3	0.5	1	2	3	4	5	6
   ln(x)						0

    (3 points)  (ii) Use the e^x button on your calculator to complete in the table (or a copy of it). 

   x	-1.792	-1.61	-1.39	-1.099	-0.693	0	0.693	1.099	1.39	1.61	1.79
   exp(x)						1

    (5 points)   (iii) Draw  the straight line y = x from [-2,-2] to [6,6] on graph paper.  Then on the same graph, use the above points to sketch graph of y = ln(x) for x in the interval [1/6, 6] and  y = exp(x) =  e^x  for x in the interval [-1.79, 1.79]. Label all curves.  
   

5. Try section 2.4 in this set of  Arithmetic Review Problems. from site Volume 2, Three Skills for Algebra.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science