If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that $$f$$ (aka there is a point where the slop of the tangent line/instantaneous rate of change equals the slope of the secant line/AROC)

$$\int_{ }^{ }\frac{1}{a^2+x^2}dx$$

$$\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C$$

Average rate of change/slope of the secant line over [a, b]

$$\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

$$\frac{d}{dx}\left(f\left(x\right)g\left(x\right)\right)=f\left(x\right)g$$

"low... d high... minus high... d low... cross the line... square what's below" $$\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right)f$$

$$\frac{d}{dx}f\left(g\left(x\right)\right)=f$$

$$\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$$

Derivative of Inverse Functions

$$\frac{d}{dx}\left(f^{-1}\left(x\right)\right)=\frac{1}{f$$

Tangent line/Linear approximation at the point (a, b)

$$y-b=f$$ (basically point-slope form)

Horizontal and Vertical Tangents

Horizontal: f'(x) = 0 (numerator = 0) Vertical: f'(x) DNE (denominator = 0)

If lim(x→a) f(x) = 0 or ±∞ and lim(x→a) g(x) = 0 or ±∞, then $$\lim_{x\rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\rightarrow a}\frac{f$$ Side note: NEVER write 0/0 or ∞/∞; write f(x) and g(x) separately

$$Let\ y=b$$ $$\lim_{x\longrightarrow\pm\infty}f\left(x\right)=b$$

$$Let\ x=a$$ $$\lim_{x\rightarrow a^+}f\left(x\right)=\pm\infty\ or\ \lim_{x\rightarrow a^-}f\left(x\right)=\pm\infty$$ or f(a) is of the form #/0

Symmetric over y-axis f(-x) = f(x)

Symmetric over origin f(-x) = -f(x)

Absolute Max/Mins on Closed Interval [a, b] (Candidates Test)

1. Candidates are critical points and endpoints 2. Find the y-value at each candidate and compare them

Intervals of Increasing/Decreasing [use brackets]

Increasing: f'(x) ≥ 0 Decreasing: f'(x) ≤ 0

1st Derivative Test: f'(x) changes from positive to negative at x = a 2nd Derivative Test: f'(a) = 0 and f"(a) < 0

1st Derivative Test: f'(x) changes from negative to positive at x = a 2nd Derivative Test: f'(a) = 0 and f"(a) > 0

1st Derivative Test for Abs. Max at x = a

f'(x) > 0 for ALL x < a and f'(x) < 0 for ALL x > a Only works if there is a SINGLE critical point.

1st Derivative Test for Absolute Min at x = a

f'(x) < 0 for ALL x < a and f'(x) > 0 for ALL x > a Only works if there is a SINGLE critical point.

Intervals of Concave Up/Down (use parenthesis)

Up: f"(x) > 0, or f'(x) is increasing Down: f"(x) < 0, or f'(x) is decreasing

Inflection Point at x = a

f"(x) changes sign at x = a, OR f'(x) changes from increasing to decreasing (or vice versa) at x = a

$$\frac{1}{2}w\left(h_1+h_2\right)$$

Limit of a Right Riemann Sum

$$\int_a^bf\left(x\right)dx=\lim_{x\rightarrow\infty}\sum_{k=1}^nf\left(a+k\Delta x\right)\Delta x$$ where $$\Delta x=\frac{b-a}{n}$$

Does the Riemann sum over/underapproximate... $$\int_a^bf\left(x\right)dx$$

Left/Right sum: is f(x) increasing or decreasing? Midpoint/Trapezoid: Is f(x) concave up or down?

AP Calculus BC Formulas & Concepts Flashcard

Student preview

117 questions

Show all answers

1.

FLASHCARD QUESTION

Front

Definition of a Limit

Back

2.

FLASHCARD QUESTION

Front

Definition of Continuity at x = a

Back

3.

FLASHCARD QUESTION

Front

Squeeze/Sandwich Theorem

Back

4.

FLASHCARD QUESTION

Front

Intermediate Value Theorem

Back

If f is continuous on [a, b] and y is a number between f(a) and f(b), then there exists a c in (a, b) such that f(c) = y (aka f(x) takes on every y-value between f(a) and f(b))

5.

FLASHCARD QUESTION

Front

Extreme Value Theorem

Back

If f is continuous on [a, b], then there exists a c and d in [a, b] such that f(c) ≥ f(x) and f(d) ≤ f(x) for all x (aka f has an absolute max value f(c) and absolute min value f(d))

6.

FLASHCARD QUESTION

Front

Mean Value Theorem

Back

7.

FLASHCARD QUESTION

Front

Back

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