ENGR 133
PROBLEM SET: SYMBOLIC PROCESSING ASSIGNMENT INSTRUCTIONS
OVERVIEW
Problem Set (PS) Assignments serve to validate comprehension of the new material introduced
in the module and serve to prepare you for upcoming quizzes.
At the completion of this Problem Set, you should be able to:
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Create symbolic expressions and manipulate them algebraically.
Obtain symbolic solutions to algebraic and transcendental equations.
Perform symbolic differentiation and integration.
Evaluate limits and series symbolically.
Plot symbolic functions.
It is highly recommended that you complete the Read and Watch items outlined in the Learn
section of the module, as well as the Quiz and Lab Assignment for the module before attempting
the Problem Set. It is important to start the Problem Set sufficiently in advance of the due date to
maximize the use of the resources available before the due date.
INSTRUCTIONS
Start MATLAB and perform all of the operations outlined in the assignment instructions. Insert
comments as appropriate to solidify your understanding and clearly communicate the concepts
presented. When complete, upload the pdf file created for the assignment no later than the due
date/time. If you are unable to create a pdf file, then upload all the files you were able to create
for the assignment.
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Review the problem-solving technique described in section 1.6 of the text. Pay close
attention to how the method is applied in the examples given.
Using the methodology presented in section 1.6, solve the problems below and publish
the results in a single pdf file. The assignment in Canvas has additional details.
Upload the published pdf file to the appropriate location in Canvas before the due date.
End-of-Chapter 11 problems:
o Problem 11.6 (straightforward application of ‘syms’, ‘solve’, ‘subs’, and ‘double’.
o Problem 11.8 (Confirm your solution with a plot. Hint: Use ‘fimplicit’)
o Problem 11.11 (A horizontal tangent line occurs when dy/dy = 0. Confirm your
solution with a plot – be sure to set the plot axes so that the solution is clear.
o Problem 11.13 (Combine the two equations to create a single equation containing S
and V, then take the derivative dS/dV. Substitute V = 30 to determine the answer
(1.0376 in2/in3)). This does not mean that your answer is incorrect – it means that
you should be diligent in including some code that will check your answer to verify if
it is correct.
o Problem 11.21 (Straightforward application of the ‘int’ and ‘fplot’ functions. See
warnings note on the previous problem.
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ENGR 133
Problem-Solving Tips
11.6 The law of cosines for a triangle states that 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 ∙ cos⁡(𝐴), where a is the
length of the side opposite the angle A, and b and c are the lengths of the other sides.
(a) Use MATLAB to solve for b.
(b) Suppose that A = 60°, a = 5 m, and c = 2 m. Determine b.
Approach
This is a straightforward application of the ‘syms’, ‘solve’, ‘subs’, and ‘double’ commands.
11.8
The equation for an ellipse centered at the origin of the Cartesian coordinates (𝑥, 𝑦) is
𝑥2 𝑦2
+
=1
𝑎2 𝑏 2
where a and b are constants that determine the shape of the ellipse.
(a) In terms of the parameter b, use MATLAB to find the points of intersection of the two
ellipses described by
𝑥2 +
𝑦2
=1
𝑏2
and
𝑥2
+ 4𝑦 2 = 1
100
(b) Evaluate the solution obtained in part (a) for the case: 𝑏⁡ = ⁡2.
Approach
This is an application of the ‘syms’, ‘solve’, ‘subs’, and ‘double’ commands. Remember that the
intersection points will be located where both ellipse equations are equal to one another.
Check your results by plotting both ellipses. Because the equations are not in the form 𝑦 = 𝑓(𝑥),
use fimplicit() to plot.
11.11 Use MATLAB to find all the values of x where the graph of 𝑦 = 4𝑥 − 5𝑥 has a
horizontal tangent line.
Approach
It is important to remember that the tangent of a line is the slope of the line at that point. Further,
a tangent that is horizontal represents a slope of zero, which can be determined by taking the
derivative of the line and then finding the roots of the derivative (i.e., the slope = zero). Plugging
the roots back into the original function will complete the determination of the (x,y) coordinates
of the points of interest.
It is useful to plot the function to confirm the solution. Recognize that the Law of Cosines only
applies to positive values of a, b, and c; therefore, if your solution yields negative values for b,
they must be rejected.
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11.13 The surface area of a sphere of radius r is 𝑆 = 4𝜋𝑟 2 . Its volume is 𝑉 = 3 𝜋𝑟 3 .
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ENGR 133
(a) Use MATLAB to find the expression for dS/dV.
(b) A spherical balloon expands as air is pumped into it. What is the rate of increase in the
balloon’s surface area with volume when its volume is 30 cubic inches
Approach
To find dS/dV, we must combine the two equations to create an equation for S as a function of V.
The commonality between the two equations is the parameter r, but in both equations, r is raised
to a power.
𝑆
For part a, first solve the two equations to isolate the r terms, giving 𝑟 2 = 4𝜋 for the first and
3𝑉
𝑟 3 = 4𝜋 for the second. Recognize that if we cube the first equation and square the second, both
equations will represent 𝑟 6 … which can then be set equal to each other, like this:
𝑆 3
3𝑉 2
𝑟6 = ( ) = ( )
4𝜋
4𝜋
Now, all we need to do is solve for S, yielding:
3
𝑆 = √(36𝜋𝑉 2 )
For part b, we are asked to find the rate of change of S at a particular value of V. This is just
asking for us to evaluate the derivative of S at a point. So, the solution is a straightforward
application of the ‘syms’, ‘diff’, ‘subs’, and ‘double’ commands.
11.21 The equation for the voltage v(t) across a capacitor as a function of time is:
1 𝑡
𝑣(𝑡) = [∫ 𝑖(𝑡)𝑑𝑡 + 𝑄0 ]
𝐶 0
where i(t) is the applied current and Q0 is the initial charge. Suppose that C = 10−7 Farads and
that Q0 = 0. If the applied current is, 𝑖(𝑡) = 0.3 + 0.1𝑒 −5𝑡 ∙ 𝑠𝑖𝑛(25𝜋𝑡), use MATLAB to
compute and plot the voltage v(t) for 0 ≤ t ≤ 7 seconds.
Approach
Begin by declaring v(t) and i(t) as symbolic functions. This will also result in t being created as a
symbolic variable. Next, initialize the values of C, Q0, and i(t). Initializing i(t) will require use
of the symbolic integration command, ‘int’. Finally, compute the value of v(t) and then create
the plot using ‘fplot’ with the range specified.
A correct solution should look like this:
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ENGR 133
Be sure to include all the elements of a proper plot!
You can verify that your solution is correct by checking the value of v(t) at t = 7 seconds.
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