Watch the video below or download and read the video notes for a demonstration, then answer the questions below.

**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. If not, then you will benefit greatly from joining the STEM Math “Magic” Course.**

**1. Let P(x) = x ^{5} + 4.1x^{4} +1.1x^{3} – 8.2x^{2} – 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) = x ^{2} + .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(x ^{2}). **

**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 + 2.7x ^{2}).**

**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 + e^{t} = 0, y(1) = 2**

[This is a Differential Equation of order 1.]

**11. x ^{2} + y^{3} – (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.**