AP Calculus AB: Essential Formulas Flashcards

Questions and Answers

What is the limit definition of derivative?

The limit definition of derivative is the limit as h approaches 0 of the difference quotient (f(x+h) - f(x)) / h.

What is the alternate definition of the derivative of f(x) at c?

The alternate definition includes finding the limit as x approaches c of (f(x) - f(c)) / (x - c).

When is f(x) continuous?

f(x) is continuous if the limit as x approaches a equals f(a) for every point a in its domain.

When is f(x) differentiable at x=c?

f(x) is differentiable at x=c when the left-hand derivative equals the right-hand derivative at c. Signup and view all the answers

What is the power rule in calculus?

The power rule states that the derivative of x^n is n*x^(n-1). Signup and view all the answers

What is the product rule?

The product rule states that the derivative of two functions multiplied together is f'(x)g(x) + f(x)g'(x). Signup and view all the answers

What is the quotient rule?

The quotient rule states that the derivative of f(x) / g(x) is (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. Signup and view all the answers

What is the derivative of sin(x)?

The derivative of sin(x) is cos(x). Signup and view all the answers

What is the derivative of cos(x)?

The derivative of cos(x) is -sin(x). Signup and view all the answers

What is the derivative of tan(x)?

The derivative of tan(x) is sec^2(x). Signup and view all the answers

What is the derivative of cot(x)?

The derivative of cot(x) is -csc^2(x). Signup and view all the answers

What is the derivative of sec(x)?

The derivative of sec(x) is sec(x)tan(x). Signup and view all the answers

What is the derivative of csc(x)?

The derivative of csc(x) is -csc(x)cot(x). Signup and view all the answers

If a function is differentiable, what can be said about its continuity?

If a function is differentiable at a point, then it is continuous at that point. Signup and view all the answers

When can't a function be differentiable?

A function can't be differentiable at points of discontinuity, corners, cusps, or vertical tangent lines. Signup and view all the answers

When does a particle move left?

A particle moves left when V(t) < 0. Signup and view all the answers

When is a particle not moving (at rest)?

A particle is not moving when v(t) = 0. Signup and view all the answers

When does a particle change direction?

A particle changes direction when v(t) changes sign. Signup and view all the answers

When does a particle speed up?

A particle speeds up when a(t) and v(t) have the same signs. Signup and view all the answers

When does a particle slow down?

A particle slows down when a(t) and v(t) have different signs. Signup and view all the answers

What is the chain rule for derivatives?

The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x))g'(x). Signup and view all the answers

What is the derivative of arcsin(x)?

The derivative of arcsin(x) is 1 / sqrt(1 - x^2). Signup and view all the answers

What is the derivative of arccos(x)?

The derivative of arccos(x) is -1 / sqrt(1 - x^2). Signup and view all the answers

What is the derivative of arctan(x)?

The derivative of arctan(x) is 1 / (1 + x^2). Signup and view all the answers

What is the derivative of arccot(x)?

The derivative of arccot(x) is -1 / (1 + x^2). Signup and view all the answers

What is the derivative of arcsec(x)?

The derivative of arcsec(x) is 1 / (|x|sqrt(x^2 - 1)). Signup and view all the answers

What is the derivative of arccsc(x)?

The derivative of arccsc(x) is -1 / (|x|sqrt(x^2 - 1)). Signup and view all the answers

What is the derivative of ln(x)?

The derivative of ln(x) is 1 / x. Signup and view all the answers

What is the derivative of log base a?

The derivative of log base a is 1 / (x ln(a)). Signup and view all the answers

What is the derivative of the natural exponential (e^x)?

The derivative of e^x is e^x. Signup and view all the answers

What is the derivative of a^x?

The derivative of a^x is a^x ln(a). Signup and view all the answers

What is the derivative of an inverse function if f(b)=a and g(b)=a?

The derivative of the inverse function is 1 / f'(g(b)). Signup and view all the answers

When does a vertical tangent occur at x=a?

A vertical tangent occurs when the limit as x approaches a of f'(x) equals positive or negative infinity. Signup and view all the answers

What is a vertical asymptote?

A vertical asymptote occurs when f(x) approaches infinity as x approaches a certain value. Signup and view all the answers

What is a horizontal asymptote?

A horizontal asymptote occurs when f(x) approaches a constant value as x approaches infinity. Signup and view all the answers

What is a horizontal tangent line?

A horizontal tangent line occurs when f'(x) = 0. Signup and view all the answers

If f'(x) > 0, what can be said about the function?

If f'(x) > 0, the function is increasing. Signup and view all the answers

If f'(x) = 0, what does it indicate?

If f'(x) = 0, the function has a horizontal tangent and may have a local extremum. Signup and view all the answers

What is the second derivative?

The second derivative is the derivative of the derivative, indicating the rate of change of the rate of change. Signup and view all the answers

Study Notes

Limit Definition of Derivative

The derivative of a function f(x) at a point c is the limit of the difference quotient as h approaches 0:
f'(c) = lim (h→0) [(f(c+h) - f(c))/h].

Alternate Definition of Derivative

Defined as the limit:
f'(c) = lim (x→c) [(f(x) - f(c))/(x - c)], providing an alternative perspective using the variable x instead of h.

Continuity of f(x)

A function f(x) is continuous at a point c if:
1. f(c) is defined,
2. lim (x→c) f(x) exists,
3. lim (x→c) f(x) = f(c).

Differentiability at x=c

For f(x) to be differentiable at x=c, the left-hand and right-hand derivatives must be equal:
f'(-)(c) = f'(+)(c).

Power Rule

Derivative of f(x) = x^n is expressed as:
f'(x) = nx^(n-1), where n is a constant.

Product Rule

For two functions u(x) and v(x), the derivative is given by:
(u·v)' = u'v + uv'.

Quotient Rule

For functions u(x) and v(x), the derivative is:
(u/v)' = (u'v - uv')/v^2.

Derivative of sin(x)

Derivative of sin(x) is:
cos(x).

Derivative of cos(x)

Derivative of cos(x) is:
-sin(x).

Derivative of tan(x)

Derivative of tan(x) is:
sec^2(x).

Derivative of cot(x)

Derivative of cot(x) is:
-csc^2(x).

Derivative of sec(x)

Derivative of sec(x) is:
sec(x)tan(x).

Derivative of csc(x)

Derivative of csc(x) is:
-csc(x)cot(x).

Differentiability and Continuity

A function must be continuous to be differentiable; this means differentiability implies continuity.

Non-Differentiable Functions

A function cannot be differentiable at points of discontinuity, corners, cusps, or vertical tangent lines.

Particle Movement Direction

A particle moves to the left when its velocity V(t) is less than 0.

Particle at Rest

A particle is stationary when its velocity V(t) equals 0.

Direction Change of a Particle

A particle changes direction when the velocity V(t) changes sign, indicating crossing the x-axis on a V(t) graph.

Particle Speed Increasing

A particle speeds up when acceleration a(t) and velocity v(t) share the same signs.

Particle Speed Decreasing

A particle slows down when acceleration a(t) and velocity v(t) have different signs.

Chain Rule for Derivatives

If y = f(g(x)), then the derivative is:
dy/dx = f'(g(x))·g'(x).

Derivative of arcsin(x)

Derivative of arcsin(x) is:
1/√(1 - x^2).

Derivative of arccos(x)

Derivative of arccos(x) is:
-1/√(1 - x^2).

Derivative of arctan(x)

Derivative of arctan(x) is:
1/(1 + x^2).

Derivative of arccot(x)

Derivative of arccot(x) is:
-1/(1 + x^2).

Derivative of arcsec(x)

Derivative of arcsec(x) is:
1/(|x|√(x^2 - 1)).

Derivative of arccsc(x)

Derivative of arccsc(x) is:
-1/(|x|√(x^2 - 1)).

Derivative of ln(x)

Derivative of ln(x) is:
1/x.

Derivative of log base a

Derivative of log_a(x) is:
1/(x ln(a)).

Derivative of natural exponential (e^x)

Derivative of e^x is:
e^x.

Derivative of a^x

Derivative of a^x is:
a^x ln(a).

Derivative of Inverse Functions

If f(b) = a and g(a) = b, then:
g'(a) = 1 / f'(b).

Vertical Tangent Line

A vertical tangent occurs at x=a if:
lim (x→a) f'(x) = ±∞ and f'(x) has a nonzero numerator over zero denominator.

Vertical Asymptote

A vertical asymptote is a line x = k where the function f(x) approaches infinity as x approaches k.

Horizontal Asymptote

A horizontal asymptote indicates that as x approaches infinity or negative infinity, f(x) approaches a constant value.

Horizontal Tangent Line

A horizontal tangent line indicates that the derivative f'(x) equals 0 at that point.

Increasing Function

If f'(x) > 0, the function f(x) is increasing at that interval.

Concave Up Function

If f''(x) > 0, the function is concave up in that interval.

Second Derivative

The second derivative, f''(x), provides information about the concavity of the function and is used in the second derivative test for local extrema.

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Test your knowledge of essential calculus formulas with these flashcards. This quiz includes key concepts like the limit definition of derivative and conditions for continuity and differentiability of functions. Perfect for AP Calculus AB students looking to reinforce their understanding before exams.