Mathematics Quiz – 1 0 12345678910 Mathematics Mathematics Quiz – 1 1 / 10 Question The Maclaurin series expansion of [math]cos(x)[/math] is: [math]sum_{n=0}^infty rac{x^n}{n!}[/math] [math]sum_{n=0}^infty (-1)^n rac{x^{2n}}{(2n)!}[/math] [math]sum_{n=0}^infty rac{x^{n+1}}{(n+1)!}[/math] [math]sum_{n=0}^infty rac{x^{2n+1}}{(2n+1)!}[/math] Explanation: The expansion is [math]sum_{n=0}^infty (-1)^n rac{x^{2n}}{(2n)!}[/math]. 2 / 10 Question What is the radius of convergence of the power series [math]sum_{n=0}^infty rac{x^n}{n!}[/math]? 0 1 2 Infinite Explanation: This is the power series of [math]e^x[/math], which converges for all [math]x[/math], hence the radius is infinite. 3 / 10 Question The solution to [math] rac{dy}{dx} = y[/math] is: [math]Ce^x[/math] [math]e^x + C[/math] [math]e^{2x}[/math] [math]Ce^{-x}[/math] Explanation: The solution is [math]y = Ce^x[/math], where [math]C[/math] is a constant. 4 / 10 Question Find the area under the curve [math]f(x) = x^2[/math] from [math]x=0[/math] to [math]x=3[/math]: 6 9 12 18 Explanation: The integral of [math]x^2[/math] over [0,3] is [math] int_0^3 x^2 dx = left[ rac{x^3}{3} ight]_0^3 = 9[/math]. 5 / 10 Question If [math]f(x) = x^2 – 2x + 1[/math], then the minimum value of [math]f(x)[/math] is: 0 1 -1 2 Explanation: [math]f(x)[/math] is a parabola opening upwards, and its minimum is at [math]x=1[/math]. Thus, [math]f(1) = 0[/math]. 6 / 10 Question The eigenvalues of [math]A = egin{bmatrix} 2 & 1 1 & 2 end{bmatrix}[/math] are: [math]1, 2[/math] [math]3, 1[/math] [math]2, 2[/math] [math]0, 1[/math] Explanation: Solving [math]det(A – lambda I) = 0[/math], we get [math]lambda = 3[/math] and [math]lambda = 1[/math]. 7 / 10 Question Solve for [math]x[/math] if [math]x^2 – 4x + 4 = 0[/math]: 2 -2 4 1 Explanation: This is a perfect square equation: [math](x-2)^2 = 0[/math], so [math]x = 2[/math]. 8 / 10 Question If [math]y = e^{3x}[/math], then [math]dy/dx[/math] is: [math]3e^{3x}[/math] [math]2e^{2x}[/math] [math]e^{x}[/math] [math]4e^{3x}[/math] Explanation: By differentiation, [math]dy/dx = 3e^{3x}[/math]. 9 / 10 Question The integral of [math]f(x) = 6x^2[/math] with respect to [math]x[/math] over [0, 2] is: 10 16 18 20 Explanation: The integral of [math]f(x) = 6x^2[/math] over [0, 2] is [math] int_0^2 6x^2 dx = [2x^3]_0^2 = 16[/math]. 10 / 10 Question [math]f(x) = x^3 – 3x + 1[/math], then the value of [math]f'(2)[/math] is: 5 9 10 11 Explanation: The derivative of [math]f(x) = x^3 – 3x + 1[/math] is [math]f'(x) = 3x^2 – 3[/math]. So, [math]f'(2) = 3(2)^2 – 3 = 9[/math]. Your score is The average score is 0% LinkedIn Facebook Twitter VKontakte Restart quiz
