Here are a dozen challenging math problems that any STEM student should be able to solve quickly and easily if you know how to use the proper tool.
If you can complete this quiz in less than an hour, you should be ready to compete with the best educated peers you will have in a good STEM University.
You may download the STEM Math Challenge with Answers or take the 12 question STEM Math Challenge below.
1. Let P(x) = x5 + 4.1x4 +1.1x3 – 8.2x2 – 27.4x – 37.5
Find the Roots of P(x), both real and complex, to three significant digits.
[This is a Pre-calculus problem that is very difficult without a modern tool.]
2. Find the relative Maximum and Minimum Points of P(x), and find the intervals where P(x) is increasing and decreasing.
[This is a Differential Calculus Problem.]
3. Find the points of inflection of P(x) and the Concavity intervals and graph P(x).
[This is a Differential Calculus Problem.]
4. Let F(x) = x2 + .5Sin(10x). Graph F(x) from x = 1 to 3.
Find the Arc Length of this graph, and find the area beneath the graph from 1 to 3.
[This is an Integral Calculus Problem.]
5. Rotate F(x) from problem #4 about the x-axis and find both the Volume and Surface Area of the Solid of Revolution.
[This is an Integral Calculus Problem.]
6. Let G(x) = Sin(x2).
Find the anti-derivative of G(x) and the Area under the graph of G(x) from x = -1.5 to 1.5.
[This is an Integral Calculus Problem that would probably not be given in a classical calculus course since there is not an anti-derivative of G(x) consisting of the standard functions, and thus one could not easily apply the Fundamental Theorem of Calculus.]
7. Find the point of intersection of the three planes:
3.1x + 4.3y – 7.6z = 5.2 , 2.7x – 3.4y + 5.1z = -6.9, -0.9x + 4.2y – 3.8z = 8.7
[This is a Linear Algebra problem you should be able to solve in about two minutes with a modern tool.]
8. Let F(x) = 1.9/(3.1 + 2.7x2).
Find a polynomial, P(x), of degree 8 which is the best approximation of F(x) at x = 0.
Plot both F(x) and P(x) and observe that P(x) is a very good approximation of F(x) from -.5<x<.5.
[This is an advanced Differential Calculus Problem that would be very time consuming classically, but very easy with a modern tool.]
9. y is a function of x.
Find the solution to the differential equation:
y’’ + y’ + y = cos(x), with initial conditions y(0) = .5, and y’(0) = .3
[This is a Differential Equation of order 2.]
10. y is a function of t. Find the solution to the differential equation:
y’ +y + et = 0, y(1) = 2
[This is a Differential Equation of order 1.]
11. x2 + y3 – (xy)2 = 0 implicitly defines three functions y of x.
What is the asymptotic behavior of these three functions as x → ∞ and x → -∞ ?
Also, graph these three functions and their asymptotes.
12. What is the length of the curve defined parametrically (sin(t),cos(3t)) from t = 0 to Pi?
Graph the curve.