Mathematics Remedial Examination Study Notes

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

A function is a set of ordered pairs where each input (x-value) is associated with exactly one output (y-value).

A line segment has a defined start and end point, while a ray extends infinitely in one direction from a starting point.

The logarithmic function is the inverse operation of exponentiation. It helps us solve equations where the unknown variable is in the exponent.

The integral of a function represents the area under its curve between two given points. It is a fundamental concept in calculus.

The natural logarithm of 2 is a constant value representing the exponent to which the mathematical constant e must be raised to obtain 2. It is denoted as ln(2).

The derivative of the integral of a function is the function itself.

The natural logarithm of 1/2 is a constant value representing the exponent to which the mathematical constant e must be raised to obtain 1/2. It is denoted as ln(1/2).

The integral of a function represents the area under the curve of the function.

The natural logarithm of 1 (ln(1)) is always equal to 0. This means that e raised to the power of 0 equals 1. In simpler terms, anything raised to the power of 0 equals 1.

The derivative of a function at a particular point gives the slope of the tangent line to the curve at that point.

A local maximum is the highest point on a specific interval, while a global maximum is the absolute highest point across the entire domain.

The derivative of a function measures the rate of change of the function with respect to its input variable. It is found by calculating the limit of the difference quotient as the change in the input approaches zero. Finding the derivative of a function allows us to analyze its slope and behavior at any point.

A convergent series has a sum that approaches a finite value, while a divergent series has a sum that either approaches infinity or doesn't exist.

The integral of a function represents the area under the curve of the function with respect to its input variable. It can be thought of as summing up infinitely many small areas under the curve. Finding the integral of a function provides insights into the accumulated change of the function over a specific interval.

Complex numbers consist of a real part and an imaginary part, expressed in the form 'a + bi', where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

The fundamental theorem of calculus establishes a fundamental relationship between differentiation and integration, stating that differentiation and integration are inverse operations. This means that evaluating the integral of a function's derivative returns the original function plus a constant.

The solution to the differential equation y' + y = 0, with initial condition y(0) = 2, is y = 2e^(-x).

The limit of f(x) = (sin(x) - x)/(x^3 - 1) as x approaches 0 is -1/3.

The slope of a line perpendicular to y = 2x + 5 is the negative reciprocal of 2, which is -1/2. The equation of a line with a slope of -1/2 can be written in the form y = -1/2x + b, where b is the y-intercept.

A function has a horizontal tangent at a point where its derivative is equal to zero. The derivative of f(x) is f'(x) = 3x^2 + 2kx + 2. Setting f'(1) = 0 gives 3 + 2k + 2 = 0, which yields k = -5/2.

The slope of the tangent line at a point on a curve is found by evaluating the derivative of the function at that point. The derivative of y = x^2 + 3x - 2 is y' = 2x + 3. At the point (1, 2), the slope is y'(1) = 2 + 3 = 5. The equation of the tangent line is y - 2 = 5(x - 1), which simplifies to y = 5x - 3.

The definite integral of a function represents the area under its graph between two points. Calculate the antiderivative of x^3 - 2x^2 + x + 1, which is (x^4)/4 - (2x^3)/3 + (x^2)/2 + x. Evaluate this antiderivative at x = 2 and x = 0 and subtract the results: [(2^4)/4 - (22^3)/3 + (2^2)/2 + 2] - [(0^4)/4 - (20^3)/3 + (0^2)/2 + 0] = 4 - 16/3 + 2 + 2 = 10/3

The standard equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Therefore, the equation of the circle with center (-2, 3) and radius 5 is (x + 2)^2 + (y - 3)^2 = 5^2, which simplifies to (x + 2)^2 + (y - 3)^2 = 25.

The normal line to a curve at a point is perpendicular to the tangent line at that point. First, find the slope of the tangent line: y' = 2x - 4, so y'(2) = 0. The slope of the normal line is the negative reciprocal of 0, which is undefined. This means the normal line is vertical and its equation is x = 2.

The derivative of ln(u) is u'/u, where u is a function of x. In this case, u = x^2 + 1, so u' = 2x. Therefore, the derivative of ln(x^2 + 1) with respect to x is (2x)/(x^2 + 1).

The equation e^(2x) - 3e^x + 2 = 0 can be factored as (e^x - 1)(e^x - 2) = 0. This gives us two possible solutions: e^x = 1 or e^x = 2. Taking the natural logarithm of both sides of each equation, we find x = 0 or x = ln(2).