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Math 150A Project.
In the prerequisite classes for calculus, we learn many rules (formulas) and gain
some intuition. Use calculus to prove or derive the following ideas learned or understood
in earlier classes. (25 points)
Now pick at least 2 of the questions below and write a paper detailing the algebra,
trigonometry or geometry involved in the problem. You will need to reference algebra,
trigonometry or geometry text books to do this. Discuss the aspects of the problem you
choose at the prerequisite level and then discuss the calculus that helped you complete the
project problem (I will give an example of this discussion in class). Your paper should be
2 pages and written in APA or MLA format. (25 points)
All aspects of this project should be typed and points will be awarded based on
correctness, completeness and presentation (neat not fancy).
ALGEBRA
1. Quadratic functions are written in the form f(x) = ax2 + bx + c. minimizing or
maximizing this function is done by finding the vertex. Use calculus to derive the
formula used to find the x-coordinate of the vertex.
2. H(t) = -16t2 + vt + s gives a function that gives the height of an object in feet
relative to time in seconds. Use calculus to derive the function where the initial
velocity of a projectile is v and initial height is s.
3. On a circle, any tangent line is perpendicular to the radius that intersects the circle
at the point of tangency. Use calculus to prove this.
4. When maximizing the area of a rectangle with perimeter P, The shape turns out to
be a square. Use calculus to prove this.
TRIGONOMETRY
1. In a triangle, when given two sides and the included angle there is a formula for
finding the area of the triangle. Find the measure of the angle that maximizes the
area of the triangle.
2. In the function f(x) = Asinx, A is called amplitude. Use calculus to prove that A
and –A are the maximum and minimum values that f(x) attains during any one of
its cycles.
GEOMETRY
1. The minimum distance between a line (y=mx+b) and a point not on the line (c,d)
has the following characteristic. The line connecting the point (c,d) with the point
on the line closest to (c,d) is perpendicular to y = mx + b (this is what you will be
proving). Use the distance formula to find the distance function between the point
and the line. Then use calculus to find the point on y = mx + b closest to (c,d) (in
terms of c, d, m, and b). Show that the line connecting (c,d) to the point on
y=mx+b closest to it is perpendicular to y = mx + b.

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