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Math 1401 Calculus 1

Evaluating Limits

Raphael Sevilla

September 12, 2014

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Representing, Evaluating, and Graphing Functions Mathematica Project

Raphael SevillaMATH 1401 Calculus 1September 12, 2014

Objectives : The project explored the representation of functions through seeing them visuallyand algebraicallyCommands: Solve; NSolve; Expand; Collect; Apart; Together; Numerator; Denominator;Factor; Plot; Solve; N; Reduce

1a. The problem asked if there were any visible x intercepts in the graph the square root of xsquared multiplied by the sin of x plus of Pi/6. In order to find out if there were any x intercepts,I used the Plot command on Mathematica and it plotted out the following function. From lookingat the graph, there seemed to be points that hit the x axis, yet not passing zero, so I concludedthat there were x intercepts.

1b. The problem asked to determine what the solve command says about x intercepts of thefunction f(x). After putting the Solve command to solve f(x), it said that their values wereapproaching zero. Using that information, I determined that the x intercepts were approachingzero.

1c. The problem asked to further enhance the view of the graph f(x) to hold the x range from -4to 4 and asked whether it had x intercepts. I plotted the following the function. I looked againwhether if there were points that hit the x axis, andthere are several points that hit it so I resolved thatthere are x intercepts.

1d. The problem asked whether there were x intercepts in different x intervals, from -6 to 3 andfrom -20 to 20. From plotting both functions, I see that there are points that hit the x axis, yet no

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points that truly intersect the function.

2a. The problem asks whether why f(x) should have the same x intercepts as g(x). Fromexamining the functions, the only difference between the two graphs is the square root. Thesquare root does not limit the zeroes because the function of g(x) only limits the negative y

values yet not the zeroes.

2b. The problem asks what is true of the x values for the function f(x). After examining the f(x)graph, these values were not existent because the square root values of negative numbers do notexist.

2c. The problem asks what the commands FindRoot within the interval of 0 to 5 and from Pi/4 to3 Pi/4. I plugged in the command of FindRoot for both intervals and the output said that bothvalues were approaching zero. In addition, I used the command Reduce into Mathematica andthat the roots are also approaching very close to zero.

3a. The problem asked to complete the square by hand. After completing the square, the valuesof h, k, a are correspondingly -1, 2, 1

3b. The problem asked to plot the following pairs of quadratics by hand on the same axis

3c. The problem asks for the effects of the a, h and k values on the graph. I plugged in severalvalues of a, h, and k in Mathematica and I examined the graphs visually. After plugging thevalues of a, I figured out that a stretches the graph. The value of h determines the horizontal shiftof the graph. Also, the value of k represents the vertical shift

3d. The problem asks to determine how their roots, their highest points, and their lowest pointsrelated. After plotting both graphs, I examined that both graphs share the same vertex. Similarly,their highest points are similar if a is positive and their lowest points are shared when a is

positive. When a is negative, the graph of g(x) has even lower points.

3e. The problem asks for the relationship between f(x) = x^2 and g(x)= (x-h)^2. After plotting both functions have the same shape, yet g(x) has a horizontal translation.

3f. The problem asks for the relationship between f(x) = x^2 and g(x)= x^2+k. I plotted in bothfunctions in Mathematica. Their roots are the same when k is zero. As k increases the highestvalue increases and the lowest value also increases.

3g. The problem asks to take the whole quadratic function and account it into the relationship between the two graphs. The lowest point of the graph g(x) is determined by the h and k values.A stretches the graph and determines its placement whether if it has positive y values ornegative x values

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