Differential Calculus
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Differential Calculus/C.Akkoc/1996

MODULE No.1

GRAPHING FUNCTIONS (I)

Construct the graphs for the following functions on PLOT . Make a hard copy of each graph on a printer. Find all X & Y intercepts algebraically and graphically; zero in on the root with the cursor (eyeballing) to get an approximation for the intercept(s). Draw all asymptotes (vertical,horizontal,oblique,CURVED) on the hard copy by hand (pencil only) using a ruler. Hand label (equations) all asymptotes. Find the locations of MAXIMUM, MINIMUM and INFLECTION points on the hard copy by moving the cursor to the said locations (eyeballing) and reading off the coordinates from the upper left hand corner of the screen. Calculate ANALYTICALLY the coordinates for maxima/minima and inflection point(s) using derivative(s). Compare with your graphical estimates. Hand label actual (exact) maxima/minima and inflection points on the hard copy. Hand label regions where the graph is concave up/down.

  1. f(x) = x^2 + 2/x
    PLOTtish: f(x)=x^2+2/x ; Window: X[-4,4],Y[-10,10]

  2. f(x) = x^3 - 9x^2 + 12x - 6
    PLOTtish: f(x)=x^3-9*x^2+12*x-6; Window: X[-2,10],Y[-60,50]
 Differential Calculus
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Differential Calculus/C.Akkoc/1996

MODULE No.2

GRAPHING FUNCTIONS (II)

Construct the graph of the following function on TWIDDLE or PLOT . Make a hard copy of the graph on a printer. Find all X & Y intercepts algebraically and graphically; zero in on the root with the cursor (eyeballing) to get an approximation for the intercept(s). Draw all asymptotes (vertical,horizontal,oblique, CURVED) on the hard copy by hand (pencil only) using a ruler. Hand label (equations) all asymptotes. Find the locations of MAXIMUM, MINIMUM and INFLECTION points on the hard copy by moving the cursor to the said locations (eyeballing) and reading off the coordinates from the upper left hand of the screen. Calculate ANALYTICALLY the coordinates for maxima/minima and inflection point(s) using derivative(s). Compare with your graphical estimates. Hand label actual (exact) maxima/minima and inflection points on the hard copy. Hand label regions where the graph is concave up/down.

f(x) = (x - 1)^2/(x - 2)^2

TWIDDLEish: f(x)=(x-1)^2/(x-2)^2; Window: X[-5,10],Y[-0.5,4]
 Differential Calculus
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Differential Calculus/C.Akkoc/1996

MODULE No.3

GRAPHING FUNCTIONS (III)

Construct the graph of the following function on TWIDDLE or PLOT . Make a hard copy of the graph on a printer. Find all X & Y intercepts algebraically and graphically; zero in on the root with the cursor (eyeballing) to get an approximation for the intercept(s). Draw all asymptotes (vertical,horizontal,oblique,CURVED) on the hard copy by hand (pencil only) using a ruler. Hand label (equations) all asymptotes. Find the locations of MAXIMUM, MINIMUM and INFLECTION points on the hard copy by moving the cursor to the said locations (eyeballing) and reading off the coordinates from the upper left hand of the screen. Calculate ANALYTICALLY the coordinates for maxima/minima and inflection point(s) using derivative(s). Compare with your graphical estimates. Hand label actual (exact) maxima/minima and inflection points on the hard copy. Hand label regions where the graph is concave up/down.

f(x) = x^(2/3) - (1/5)*x^(5/3)

TWIDDLE/PLOT will not plot negative numbers (base) with non-integer powers; thus it will not plot on x<0. Go to the software PLOT and use the PARAMETRIC FORM to by-pass this problem. The given function f(x) in parametric form will be:

x(t) = t^3 , y(t) = t^2 - (1/5)*t^5 ; t in [-8,2]

PLOTtish: x(t)=t^3, y(t)=t^2-(1/5)*t^5; Range[-8,2]; Window: X[-4,6],Y[-1,4]
All analytical calculations may be done in non-parametric form.
 Differential Calculus
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Differential Calculus/C.Akkoc/1996

MODULE No.4

PARAMETRIC REPRESENTATION of FUNCTIONS (I)

Given the functional relationship in parametric form:

x(t) = t^2 , y(t) = (t^3)/3 - t ; t real

  1. Graph the given function on PLOT in PARAMETRIC FORM and make a hard copy on a printer. Mark the direction of "flow" on the hard copy by hand for "t" advancing in the positive direction. Mark the values of "t" on key points on the hard copy.
  2. Find all points on the hard copy where the curve crosses itself and mark the corresponding t-values on the graph.
  3. Calculate the slope of the tangent line at t= -sqrt(3), 0, 1, +sqrt(3) and draw the tangent line by hand on the hard copy at the corresponding points on the graph.
  4. Calculate the second derivative d^2y/dx^2 at points t=-sqrt(3), 0, 1, +sqrt(3) and write the values on the graph next to the indicated points.
 Differential Calculus
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Differential Calculus/C.Akkoc/1996

MODULE No.5

PARAMETRIC REPRESENTATION of FUNCTIONS (II)


Given the algebraic relationship:

y=3^(y/x)


  1. Determine if y(x) is a proper function; determine if x(y) is a proper function. You may need to change the form of the given equation by using the NATURAL LOGARITHM FUNCTION. Please justify using the definition of a function.
  2. Try to graph y(x) and/or x(y) on PLOT on separate screens and make a hard copy of each on a printer. Use a window X[-1,10],Y[-10,10]. Label all graphs by hand on the hard copy.
  3. Transform the given non-parametric equation into a pair of parametric equations in the form x=x(t), y=y(t).
  4. Graph the parametric equations found in Step-3 on PLOT using the conventional layout; Y-axis vertical, X-axis horizontal. Use a window X[-8,10],Y[-1,10].
  5. Find all asymptotes, vertical, horizontal, oblique, if they exist. Make use of both the parametric and non-parametric representations for this purpose.
  6. Find all points on the graph of y(x) where the tangent to the graph is a VERTICAL line. You may use the non-parametric form dy/dx or dx/dy or the parametric form dy/dx=dy/dt:dx/dt or dx/dy=dx/dt:dy/dt.
    Note: d/d(something) of ln(something)=1/something.
  7. Determine and describe the mapping from the real line (t-axis) onto the x-y plane characterized by the parametric representation constructed in Step-3. Show on the hard copy in Step-4 how different segments of the real line are mapped into various parts of the parametric graph by using colors, character code,...
 Differential Calculus
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Differential Calculus/C.Akkoc/1996

MODULE No.6

GEOMETRIC CONNECTIONS BETWEEN: f(x), f'(x), f"(x)


Given the Polynomial:
f(x) = x^3 - 9x^2 + 12x - 6

  1. Calculate f'(x) and f"(x) algebraically.
  2. Graph f(x),f'(x),f"(x) on PLOT (all three graphs on the same screen) and make a hard copy on a printer. Use a window X[-2,10], Y[-50,45]. Label all three graphs f(x),f'(x),f"(x) by hand on the hard copy.
  3. Find all local maxima, minima, and inflection points for the given function by analysis and locate (mark coordinates) such points on the hard copy.
  4. Show ALL ANALYTICAL CONNECTIONS pertaining to key points by marking relevant points on the hard copy between the graphs of f(x), f'(x), and f"(x). Examples of such connections would be between:

    f(max/min) and f'(x-intercepts);

    f(inflection points) and f'(max/min) and f"(x-intercepts)

  5. Show ALL ANALYTICAL CONNECTIONS pertaining to intervals by marking relevant intervals on the hard copy between the graphs of f(x), f'(x), and f"(x). Examples of such connections would be between:

    f(increasing/decreasing) and f'(positive/negative);

    f(concave up/down) and f"(positive/negative).