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 Flashcards

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

Access all questions and much more by creating a free account

Create resources

Host any resource

Get auto-graded reports

Continue with Google

Continue with Email

Continue with Classlink

Continue with Clever

or continue with

Microsoft

Apple

Others

Already have an account?

Similar Resources on Wayground

92 questions

Destrezas básicas

Flashcard

•

6th - 8th Grade

76 questions

Herbal Medicine Flashcards

Flashcard

•

KG

74 questions

Civil Engineering Law - Republic Act No. 544

Flashcard

•

KG

70 questions

Conjugaciones del subjuntivo en español

Flashcard

•

Professional Development

111 questions

G9-U10-Planet Earth

Flashcard

•

9th Grade - University

105 questions

T5Condicionamiento Instrumental y Programas de Reforzamiento

Flashcard

•

KG

100 questions

Como utilizar os laboratórios de aprendizagem da Academy AWS

Flashcard

•

University

131 questions

Irodalmi fogalmak p-től s-ig

Flashcard

•

11th Grade

Popular Resources on Wayground

7 questions

History of Valentine's Day

Interactive video

•

4th Grade

15 questions

Fractions on a Number Line

Quiz

•

3rd Grade

20 questions

Equivalent Fractions

Quiz

•

3rd Grade

25 questions

Multiplication Facts

Quiz

•

5th Grade

$fractions$

22 questions

fractions

Quiz

•

3rd Grade

15 questions

Valentine's Day Trivia

Quiz

•

3rd Grade

20 questions

Main Idea and Details

Quiz

•

5th Grade

20 questions

Context Clues

Quiz

•

6th Grade

Discover more resources for Mathematics

20 questions

Graphing Inequalities on a Number Line

Quiz

•

6th - 9th Grade

20 questions

Exponent Properties

Quiz

•

9th Grade

15 questions

Combine Like Terms and Distributive Property

Quiz

•

8th - 9th Grade

20 questions

Function or Not a Function

Quiz

•

8th - 9th Grade

10 questions

Factor Quadratic Expressions with Various Coefficients

Quiz

•

9th - 12th Grade

10 questions

Elijah McCoy: Innovations and Impact in Black History

Interactive video

•

6th - 10th Grade

10 questions

Evaluating Piecewise Functions Practice

Quiz

•

11th Grade

21 questions

Factoring Trinomials (a=1)

Quiz

•

9th Grade