MathProblemsBank - bank of mathematical problems and solutions

MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: Find the domain of convergence of the functional series: \[ \sum_{n=1}^{\infty}\left(1-4 x^{2}\right)^{n} \text {. } \]

2.9.3 Functional sequences and series

1.02 $

Problem: Examine the indicated series with positive terms for convergence. 1. \( \sum_{n=1}^{\infty} \frac{n^{n}}{(n+1) !} \), 2. \( \sum_{n=1}^{\infty}\left(\frac{1}{\ln (n+1)}\right)^{2 n} \).

2.10.1 Number series

1.27 $

Problem: Examine the alternating series for convergence and absolute convergence: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(2 n+1)}{n} \text {. } \]

2.10.2 Number series

0.51 $

Problem: With an accuracy of \( \varepsilon=0.001 \) find the partial sum \( S \) of the series \[ S=\sum_{n=1}^{\infty}\left(\frac{4 n+7}{6 n+5}\right)^{n^{2}} \cdot / \]

2.10.3 Number series

3.82 $

Problem: Examine the function for an extremum: \[ z=y \sqrt{x}-y^{2}-x+6 y . \]

2.11.1 Function extrema

1.27 $

Problem: Determine the largest and smallest values of the function \( z=x^{2}-y^{2} \) in the domain \( x^{2}+y^{2} \leq 1 \).

2.11.2 Function extrema

1.27 $

Problem: Determine the largest and smallest values of a function \( f(x)=2 x^{2}+\frac{2}{x} \) in \( \left[\frac{1}{2} ; 1\right] \).

2.11.3 Function extrema

1.27 $

Problem: Find the points of local extrema of the function \[ z=x^{3}-y^{3}-3 x y . \]

2.11.4 Function extrema

1.27 $

Problem: The vertex of the parabola coincides with one of the foci of the hyperbola \( 9 x^{2}-16 y^{2}=144 \). Find the equation of the parabola if it is known that it's directrix passes through the points \( (-4 ;-3) \) and \( (-4 ; 3) \).

3.1.1 Curves of the 2-nd order

2.04 $

Problem: Find the equation of the parabola the branches of which lie in the half-plane \( x \leq 0 \), the line \( y+4=0 \) is the axis of symmetry, and the parabola intersects the axis \( O X \) at the point \( (-5 ; 0) \).

3.1.2 Curves of the 2-nd order

2.04 $

Problem: Given the coordinates of the vertices of the triangle \( A B C-A(1,-6), B(3,4), C(-3,3) \). Find: 1.The equation of the height of \( \mathrm{CH} \) and its length; 2. The equation of the median of \( A M \) and its length; 3.The angle formed by the height of \( \mathrm{CH} \) and the median of \( A M \).

3.2.1 Lines on a plane

2.55 $

Problem: Find the type of the curve \( \Gamma x^{2}+\mathrm{H} x y-\mathrm{H} y^{2}- \) \( -4 x+\mathrm{H} y=\mathrm{H} \), where \( \Gamma=\mathrm{H}=2 \).

3.1.4 Curves of the 2-nd order

2.55 $

Problem: Determine the type of the surface of the 2nd order, bringing the equation to the canonical form: \[ x^{2}+5 y^{2}+z^{2}+2 x y+6 x z+2 y z-6=0 \text {. } \]

3.5.1 Surfaces of the 2-nd order

2.55 $

Problem: Determine the type of the second-order curve given by the equation \( x^{2}+5 y^{2}+6 x+20 y+4=0 \), reducing the equation to canonical form. Find the main characteristics of the curve.

3.1.3 Curves of the 2-nd order

2.55 $

Problem: Find the equation of the tangent plane and the normal at a given point on the surface: \[ 1+\sqrt{x^{2}+y^{2}+z^{2}}=x+y+z, \quad M(2 ; 3 ; 6) . \]

3.4.1 Tangents and normals

2.55 $

‹
1
2
...
10
...
116
117
›