APPENDIX C SELECTED TOPICS OF MATHEMATICS

According to the principles of plane geometry, what is the area of a triangle with a base 'b' and height 'h'?

What is the formula for the volume of a sphere with radius 'r'?

In algebra, for a quadratic equation ax squared + bx + c = 0 to have real roots, what condition must be met?

What is the equation of a circle with radius 'r' centered at the coordinates (a, b)?

What is the trigonometric identity for sin squared theta + cos squared theta?

For two vectors P and Q, what is the result of the dot product P dot Q if the vectors are perpendicular?

What is the derivative of sin(x) with respect to x?

What is the result of the integral of (1/x) dx?

In Newton's method for solving intractable equations of the form f(x) = 0, what is the formula to find the next estimate, x_k+1, from the current estimate, x_k?

What is a necessary condition for using Simpson's rule for numerical integration, as described in the appendix?

What is the circumference of a circle with a radius of 10 units, according to the formula in Appendix C/2?

What is the volume of a right-circular cone with a radius 'r' of 3 units and a height 'h' of 10 units?

According to the Algebra section (C/4), what is the value of log(1)?

What is the value of the 2nd order determinant with rows [3, 2] and [4, 5]?

In trigonometry, which quadrant has positive values for tan(theta) and negative values for both sin(theta) and cos(theta)?

For a vector V with components V_x, V_y, and V_z, what is the formula for its scalar magnitude, |V|?

What is the dot product of the unit vectors i dot j?

What is the cross product of the unit vectors i x j?

What is the series expansion for cos(x) according to Appendix C/8?

What is the derivative of a product of two functions, d(uv)/dx?

What is the result of the integral of cos(x) dx?

In Euler's method for numerical integration of dy/dt = f(t), what is the formula to find the next value, y_k+1, from the current value, y_k?

What is the relationship between two pairs of intersecting lines where the lines of one pair are, respectively, perpendicular to the lines of the other pair?

What is the key property of any triangle inscribed within a semicircle?

What is the formula for the lateral area of a right-circular cone with radius 'r' and slant height 'L'?

According to the Algebra section (C/4), what is the relationship for log(a/b)?

What is the equation for an ellipse centered at the origin with semi-axes 'a' and 'b'?

What is the Law of Sines for a triangle with sides a, b and opposite angles A, B?

The derivative of a vector V's direction cosines (l, m, n) with respect to time must satisfy which condition?

What is the derivative of tanh(x) with respect to x?

In a cubic equation of the form x cubed = Ax + B, where p = A/3 and q = B/2, what is the condition for the case of one real root and two imaginary roots?

What does the triple scalar product (P x Q) dot R represent geometrically?

What is the expansion for the triple vector product P x (Q x R)?

For the Fourier series expansion of a function f(x) over the interval -l < x < l, what is the formula for the coefficient b_n?

What is the indefinite integral of tanh(x) dx?

What is the indefinite integral of tan(x) dx?

In a cubic equation x cubed + ax squared + bx + c = 0, what substitution is used to transform it into the form x_0 cubed = A*x_0 + B?

According to the appendix, which numerical integration method involves approximating the function with a parabola passing through three successive points?

What is the value of the cross product of any unit vector with itself, such as i x i?

The miscellaneous relation '1 + tan squared theta' is equal to what?

The derivative of the vector dot product d(P dot Q)/dt is given by which expression?

In the series expansions, what is the range of convergence for the series (1 + x) to the power of n?

What is the result of the integral of sec(x) dx?

According to Appendix C/2, for two similar triangles, if x and b are corresponding bases and y and h are corresponding heights, what is the relationship between them?

What is the law of cosines for a triangle to find the side 'c' given sides 'a', 'b', and the angle 'C' opposite to side 'c'?

What is the sum of the interior angles (theta_1, theta_2, theta_3) of any triangle?

What is the integral of x^n dx, for n not equal to -1?

In Newton's method, what is a key reason to be cautious if the derivative f'(x_k) is close to zero?

What is the volume of a pyramid or cone with a base area 'B' and height 'h'?

What is the equation of a straight line with y-intercept 'a' and slope 'm'?

What is the commutative law for the vector dot product of two vectors P and Q?