AP Calculus AB Bible by Doug Graham

What is the main focus when evaluating limits?

We check around the point that we are approaching, not at the point itself.

What must be checked when evaluating limits at breaking points?

Both the left-hand limit and the right-hand limit must be checked.

What are breaking points in the context of limits?

Points on the graph that are undefined or where the graph is split into pieces, such as asymptotes, radicals, holes, and piece-wise functions.

What happens if the left-hand and right-hand limits disagree?

The limit does not exist (DNE) at that point.

What happens if the left-hand and right-hand limits agree?

The limit exists at that point as that value.

What is the limit of a function as x approaches 0 from the positive side?

The limit approaches infinity (∞).

What is the limit of a function as x approaches 0 from the negative side?

The limit approaches negative infinity (−∞).

What is the limit of a function as x approaches infinity?

The limit approaches 0.

How do you evaluate limits at non-breaking points?

You simply plug in the value.

What is the first step when dealing with holes in the graph?

Factor and cancel or multiply by the conjugate and cancel, then plug in the value.

What is the limit of (x^2 + 3x - 10)/(x - 2) as x approaches 2?

The limit is 7.

What must be checked when evaluating limits involving radicals?

You must first check that the limit exists on the side(s) you are checking.

What are the two possible outcomes when checking limits at a breaking point for a radical?

1) 0 if the limit works from the side being checked; 2) DNE if the limit does not work from that side.

What is the limit of (3 - x) as x approaches 3 from the left?

The limit is 0.

What is the limit of (5 - x) as x approaches 5 from the right?

The limit does not exist (DNE).

What is the limit of (x + 2) as x approaches -2 from both sides?

The limit is 0 from the left and DNE from the right, so both sides don't agree.

What is the significance of the Mean Value Theorem in calculus?

It states that for a continuous function over a closed interval, there exists at least one point where the derivative equals the average rate of change over that interval.

What is the purpose of the Fundamental Theorem of Calculus?

It connects differentiation and integration, establishing that differentiation and integration are inverse processes.

What are the applications of integrals in calculus?

Integrals can be used to find area, volume, and sums.

What is Newton's method used for?

It is used for finding successively better approximations to the roots (or zeroes) of a real-valued function.

What are the properties of logarithms that are important in calculus?

They include the product, quotient, and power rules for simplifying logarithmic expressions.

What does DNE stand for in limit notation?

DNE stands for 'Does Not Exist'.

What are the three possible outcomes when checking limits at a breaking point?

1) ∞ if the limit approaches positive infinity, 2) -∞ if it approaches negative infinity, 3) DNE if the limits from both sides do not agree.

What is the limit as x approaches 5 from the left of the function 3/(5-x)?

The limit is ∞, as checking values close to 5 from the left gives a positive answer.

How do you determine the limit as x approaches -8 for the function 7/(x+8)?

The limit is DNE because checking values close to -8 from both sides yields a negative answer on one side and a positive answer on the other.

What is the limit as x approaches -3 from the right for the function 3x/(x+3)?

The limit is -∞, as checking values close to -3 from the right gives a negative answer.

What is the limit of sin(x)/x as x approaches 0?

The limit is 1.

What is the limit of (1-cos(x))/x as x approaches 0?

The limit is 0.

What is the limit of tan(x) as x approaches 0?

The limit is 1.

What is the limit of sin(ax)/(bx) as x approaches 0?

The limit is a/b.

What is the limit of 6/(x-7) as x approaches infinity?

The limit is 0.

What happens to the limit if the denominator has a larger power than the numerator?

The limit approaches 0.

What happens to the limit if the numerator has a larger power than the denominator?

The limit approaches ∞ or -∞ depending on the sign.

What is the limit of (3-5x^2)/(9-x^3) as x approaches infinity?

The limit is -5.

What is a vertical asymptote?

A vertical asymptote occurs at the value that makes only the denominator equal to 0.

What is a hole in a function?

A hole occurs at points where both the numerator and denominator equal 0 at the same time.

What is the limit of 1/(6-x) as x approaches infinity?

The limit is 0.

What is the limit of (2x-5)/(2x^5+3) as x approaches negative infinity?

The limit is 0.

What is the limit of 5 tan(3x)/(6 sin(x) cos(x)) as x approaches 0?

The limit is 6/5.

What are the breaking points in the piecewise function f(x) = {3-x for x < -3, 2x + 1 for -3 ≤ x < 4, 9 for x ≥ 4}?

The breaking points are -3 and 4.

What is the limit of f(x) as x approaches -3 from the right?

The limit is -5.

What is the limit of f(x) as x approaches 4 from the left?

The limit is 9.

What is the limit of f(x) as x approaches -3?

The limit is DNE because the left and right limits do not agree.

What is the limit of f(x) as x approaches 7 from the right?

The limit is 9.

What is the condition for a function to be continuous at point 'a'?

A function is continuous at 'a' if lim x→−a−f(x) = lim x→−a+f(x) = f(a).

What happens if f(a) is not equal to either one-sided limit?

If f(a) is not equal to either one-sided limit, then the function is not continuous (discontinuous) at 'a'.

What is the definition of the derivative at a point 'a'?

The derivative at point 'a' is defined as f'(a) = lim h→0 [f(a + h) - f(a)] / h.

What does the derivative of a function represent geometrically?

The derivative of a function finds the slope of the tangent line to the curve.

What is the formula for the derivative of a function f(x) at all points?

The derivative at all points is f'(x) = lim h→0 [f(x + h) - f(x)] / h.

What is the relationship between the secant line and the tangent line as h approaches 0?

As h approaches 0, the secant line becomes the tangent line at the point.

What is the formula for finding the slope of the secant line?

The slope of the secant line is given by (f(x + h) - f(x)) / (x + h - x).

What is the equation of the tangent line in point-slope form?

The equation of a line in point-slope form is y - y1 = m(x - x1), where m is the slope.

How do you find the equation of the tangent line at x = 3 for f(x) = 5x^2?

First, find f(3) = 45 and f'(3) = 30. Then, the tangent line is y - 45 = 30(x - 3).

What is the equation of the normal line to a curve?

The normal line is perpendicular to the tangent line and can be expressed as y - y1 = -1/m(x - x1), where m is the slope of the tangent line.

What is the derivative of sin(x)?

The derivative of sin(x) is cos(x).

What is the derivative of 3x^4?

The derivative of 3x^4 is 12x^3.

What does the limit lim h→0 [f(x + h) - f(x)] / h represent?

It represents the definition of the derivative of the function at point x.

What is the significance of the limit process in finding derivatives?

The limit process allows us to find the instantaneous rate of change of the function at a specific point.

What is the result of the derivative of a constant function?

The derivative of a constant function is zero.

How do you determine if a function is discontinuous at a point?

A function is discontinuous at a point if any of the limits or the function value at that point do not match.

What is the derivative of f(x) = 4x^3?

The derivative of f(x) = 4x^3 is f'(x) = 12x^2.

What is the slope of the tangent line for f(x) = 3x^2?

The slope of the tangent line for f(x) = 3x^2 is f'(x) = 6x.

What is the derivative of a power function x^n?

The derivative of x^n is n*x^(n-1).

What is the limit notation for continuity at a point 'a'?

For continuity at 'a', it is required that lim x→a f(x) = f(a).

What is the derivative of the function f(x) = 5x^2 at x = 3?

The derivative at x = 3 is f'(3) = 30.

What is the limit as h approaches 0 for the expression (5/(2+h))?

The limit is 5/2.

What is the limit as h approaches 0 for the expression (1/(1+h))?

The limit is 1.

What is the derivative of the function y = 5x^3?

The derivative is 15x^2.

What is the derivative of the function y = x^4?

The derivative is 4x^3.

What is the Power Rule for derivatives?

If y = x^n, then the derivative is y' = nx^(n-1).

Provide an example of the Power Rule with y = 2x^5.

The derivative is y' = 10x^4.

Provide an example of the Power Rule with y = 5/x.

The derivative is y' = -5x^(-2).

What is the Product Rule for derivatives?

If y = f(x) * g(x), then y' = f'(x)g(x) + g'(x)f(x).

Provide an example of the Product Rule with y = x^2 sin(x).

The derivative is y' = 2x sin(x) + x^2 cos(x).

What is the Quotient Rule for derivatives?

If y = f(x)/g(x), then y' = (f'(x)g(x) - g'(x)f(x))/g(x)^2.

Provide an example of the Quotient Rule with y = sin(x)/x^3.

The derivative is y' = (cos(x)x^3 - 3x^2sin(x))/x^6.

What is the Chain Rule for derivatives?

If y = f(g(x)), then y' = f'(g(x))g'(x).

Provide an example of the Chain Rule with y = (x^2 + 1)^3.

The derivative is y' = 3(x^2 + 1)^2 * 2x = 6x(x^2 + 1)^2.

What is implicit differentiation?

It involves differentiating both sides of an equation with respect to x, treating y as a function of x.

Provide an example of implicit differentiation with x^2y + y^3 + x^2 = 5.

The derivative is 2xy + x^2(dy/dx) + 3y^2(dy/dx) + 2x = 0.

What is the derivative of sin(x)?

The derivative is cos(x).

What is the derivative of cos(x)?

The derivative is -sin(x).

What is the derivative of tan(x)?

The derivative is sec^2(x).

What is the derivative of ln(f(x))?

The derivative is (1/f(x)) * f'(x).

Provide an example of the derivative of ln(x^2 + 1).

The derivative is (2x)/(x^2 + 1).

What is the derivative of a constant multiplied by a function, y = a f(x)?

The derivative is y' = a f'(x).

What is the alternate method for differentiating y = f(x)g(x)?

Take the natural logarithm of both sides, then differentiate.

What is the derivative of y = x sin(x)?

y' = e^(sin(x) ln(x)) * cos(x) * ln(x) + sin(x) / x.

What is the derivative of y = arcsin(f(x))?

y' = 1 / sqrt(1 - (f(x))^2) * f'(x).

What is the derivative of y = arctan(f(x))?

y' = 1 / (1 + (f(x))^2) * f'(x).

What is the derivative of y = arcsec(f(x))?

y' = 1 / (|f(x)| * sqrt((f(x))^2 - 1)) * f'(x).

What is the formula for the volume of a sphere?

V = (4/3)πr^3.

What is the rate of change of volume for a sphere?

dV/dt = 4πr^2 (dr/dt).

What is the formula for the surface area of a sphere?

A = 4πr^2.

What is the rate of change of surface area for a sphere?

dA/dt = 8πr (dr/dt).

What is the formula for the area of a circle?

A = πr^2.

What is the rate of change of area for a circle?

dA/dt = 2πr (dr/dt).

What is the formula for the circumference of a circle?

C = 2πr.

What is the rate of change of circumference for a circle?

dC/dt = 2π (dr/dt).

What is the formula for the volume of a cylinder?

V = πr^2h.