Calculus Derivatives and Integrals Flashcards
Free calculus flashcards for derivative rules, the derivatives of common functions, basic integrals, limits and the major theorems. Flip, mark what you know and drill the rules you miss before your AP or college calculus exam.
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Limits
| Question | Answer |
|---|---|
| Definition of the limit of f(x) as x approaches a? | The value f(x) approaches as x gets arbitrarily close to a, from both sides. |
| lim (x→0) sin x / x = ? | 1 |
| lim (x→0) (1 − cos x) / x = ? | 0 |
| When is f continuous at x = a? | f(a) exists, the limit as x→a exists, and the two are equal. |
| What is L’Hôpital’s rule? | If the limit gives 0/0 or ∞/∞, then lim f/g = lim f’/g’ (when that limit exists). |
| lim (x→∞) 1/x = ? | 0 |
| Intermediate Value Theorem? | If f is continuous on [a, b] and N is between f(a) and f(b), then f(c) = N for some c in (a, b). |
Derivative rules
| Question | Answer |
|---|---|
| Limit definition of the derivative? | f'(x) = lim (h→0) [f(x + h) − f(x)] / h |
| Power rule? | d/dx xⁿ = n xⁿ⁻¹ |
| Constant multiple rule? | d/dx [c f(x)] = c f'(x) |
| Sum rule? | (f + g)’ = f’ + g’ |
| Product rule? | (fg)’ = f’g + fg’ |
| Quotient rule? | (f/g)’ = (f’g − fg’) / g² |
| Chain rule? | d/dx f(g(x)) = f'(g(x)) · g'(x) |
| Derivative of a constant? | 0 |
| What does f'(a) represent geometrically? | The slope of the tangent line to f at x = a. |
| Equation of the tangent line at x = a? | y − f(a) = f'(a)(x − a) |
Common derivatives
| Question | Answer |
|---|---|
| d/dx sin x? | cos x |
| d/dx cos x? | −sin x |
| d/dx tan x? | sec² x |
| d/dx sec x? | sec x tan x |
| d/dx csc x? | −csc x cot x |
| d/dx cot x? | −csc² x |
| d/dx eˣ? | eˣ |
| d/dx aˣ? | aˣ ln a |
| d/dx ln x? | 1/x |
| d/dx logₐ x? | 1 / (x ln a) |
| d/dx arcsin x? | 1 / √(1 − x²) |
| d/dx arccos x? | −1 / √(1 − x²) |
| d/dx arctan x? | 1 / (1 + x²) |
| d/dx √x? | 1 / (2√x) |
Applications of derivatives
| Question | Answer |
|---|---|
| How do you find critical points? | Solve f'(x) = 0 or find where f'(x) is undefined. |
| First derivative test? | If f’ changes from + to − at c, f has a local max; from − to +, a local min. |
| Second derivative test? | If f'(c) = 0 and f”(c) > 0, local min; if f”(c) < 0, local max. |
| What does f”(x) > 0 tell you? | f is concave up. |
| What is an inflection point? | A point where the concavity changes, usually where f” changes sign. |
| Mean Value Theorem? | If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = [f(b) − f(a)] / (b − a) for some c. |
| How are position, velocity and acceleration related? | Velocity is the derivative of position; acceleration is the derivative of velocity. |
Basic integrals
| Question | Answer |
|---|---|
| ∫ xⁿ dx (n ≠ −1)? | xⁿ⁺¹ / (n + 1) + C |
| ∫ 1/x dx? | ln|x| + C |
| ∫ eˣ dx? | eˣ + C |
| ∫ aˣ dx? | aˣ / ln a + C |
| ∫ sin x dx? | −cos x + C |
| ∫ cos x dx? | sin x + C |
| ∫ sec² x dx? | tan x + C |
| ∫ 1 / (1 + x²) dx? | arctan x + C |
| ∫ 1 / √(1 − x²) dx? | arcsin x + C |
| ∫ k dx? | kx + C |
| Why do indefinite integrals have + C? | Many functions share the same derivative, differing by a constant. |
Integration theory
| Question | Answer |
|---|---|
| Fundamental Theorem of Calculus, Part 1? | If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x). |
| Fundamental Theorem of Calculus, Part 2? | ∫ₐᵇ f(x) dx = F(b) − F(a), where F’ = f. |
| Integration by parts? | ∫ u dv = uv − ∫ v du |
| What is u-substitution? | Let u = g(x), du = g'(x) dx to turn ∫ f(g(x)) g'(x) dx into ∫ f(u) du. |
| Average value of f on [a, b]? | (1 / (b − a)) ∫ₐᵇ f(x) dx |
| Area between curves y = f(x) above y = g(x)? | ∫ₐᵇ [f(x) − g(x)] dx |
| Volume by the disk method (about the x-axis)? | V = π ∫ₐᵇ [f(x)]² dx |
Frequently asked questions
What derivative rules are most important to know?
The power, product, quotient and chain rules, plus the derivatives of sin, cos, tan, eˣ and ln x, cover most first-semester calculus.
How are derivatives and integrals related?
They are inverse operations. The Fundamental Theorem of Calculus says that differentiating an accumulation function gives back the original function.
Are these cards enough for AP Calculus?
They cover the core rules. AP Calculus also expects you to apply them to words, graphs and tables, so pair the cards with practice problems.
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