Formulas summary calculus derivative and integration PDF

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the summary of formulas on integration and differentiation, the collection of formulas that is good to memory and to use during the test...

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FORMULAS AND THEOREMS 1. Quadratic Formula: ax2 + bx + c = 0 (a ≠ 0)

2. Distance Formula:

3. Equation of a Circle: x2 + y2 = r2 center at (0, 0) and radius = r. 4. Equation of an Ellipse: center at (0, 0).

center at (h, k). 5. Area and Volume Formulas:

6. Special Angles:

7. Double Angles:

8. Pythagorean Identities: • sin2 θ + cos2 θ = 1 • 1 + tan2 θ = sec2θ • 1 + cot2 θ = csc2 θ 9. Limits:

10. L’Hôpital’s Rule for Indeterminate Forms Let lim represent one of the limits: . Suppose f(x) and g(x) are differentiable, and g′(x) ≠ 0 near c, except possibly at c, and suppose lim f(x) = 0 and lim g(x) = 0, then the lim

is an indeterminate form of

the type . Also, if lim f(x) = ±∞ and lim g(x) = ±∞, then the lim an indeterminate form of the type

. In both cases,

L’Hôpital’s Rule states that

and .

11. Rules of Differentiation: a. Definition of the Derivative of a Function:

b. Power Rule: c. Sum & Difference Rules:

d. Product Rule:

e. Quotient Rule:

Summary of Sum, Difference, Product, and Quotient Rules:

,

is

f. Chain Rule:

12. Inverse Function and Derivatives:

13. Differentiation and Integration Formulas: Integration Rules: a. b. c. d.

Differentiation Formulas:

a. b. c. d. e. f. g. h. i. j. k. l. m. n.

o. Integration Formulas: a. b. c. d. e. f. g. h. i. j. k. l.

m. n. o. More Integration Formulas: a.

b.

c. d. e. f. g.

h.

i.

Note: After evaluating an integral, always check the result by taking the derivative of the answer (i.e., taking the derivative of the antiderivative). 14. The Fundamental Theorems of Calculus:

Where F′(x)= f(x).

15. Trapezoidal Approximation:

16. Average Value of a Function:

17. Mean Value Theorem: for some c in (a, b). Mean Value Theorem for Integrals: for some c in (a, b). 18. Area Bounded by 2 Curves:

where f(x) ≥ g(x). 19. Volume of a Solid with Known Cross Section:

where A(x) is the cross section. 20. Disc Method: where f(x) = radius. 21. Using the Washer Method:

22. Distance Traveled Formulas:

23. Business Formulas::

P′(x), R′(x), C′(x) are the instantaneous rates of change of profit, revenue, and cost respectively. 24. Exponential Growth/Decay Formulas: . 25. Logistic Growth Models:

26. Integration by Parts:

Note: When matching u and dv, begin with u and follow the order of the acronym LIPET (Logarithmic, Inverse Trigonometric, Polynomial,

Exponential, and Trigonometric functions). 27. Derivatives of Parametric Functions:

28. Vector Functions: Given r (t) = f (t) i + g (t) j :

29. Arc Length of a Curve: (a) (b) Parametric Equations:

(c) Polar Equations:

30. Polar Curves: (a) Slope of r = f (θ) at (r,θ)

or written as

(b) Given r = f (θ) and α ≤ θ ≤ β, the area of the region between the curve, the origin, θ = α and θ = β: or

.

(c) Area between two Polar Curves: Given r1 = f (θ) and r2 = g (θ), 0 ≤ r1

≤ r2 and α ≤ θ ≤ β, the area between r1 and r2:

31. Series and Convergence: (a) Geometric Series:

if |r| > 1, series diverges; if |r| > 1, series converges and the sum =

.

(Partial sum of the first n terms: series.) (b) p- Series:

if p > 1, series converges; if 0 < p ≤ 1, series diverges. (c) Alternating Series:

for all geometric

… + (– 1)k ak+ …, where ak > 0 for all ks. Series converges if (1) a1 ≥ a2 ≥ a3 … ≥ ak ≥ … and (2)

.

(Note: Both conditions must be satisfied before the series converges.) Error Approximation: If S = sum of an alternating series, and Sn = partial sum of n terms, then |error| = |S – Sn| ≤ an+1. (d) Harmonic Series: diverges. Alternating Harmonic Series:

32. Convergence Tests for Series: (a) Divergence Test: Given a series

, if

then the series diverges.

(b) Ratio Test for Absolute Convergence: Given

where ak ≠ 0 for all ks and let

, then the

series (1) converges absolutely if p < 1; (2) diverges if p > 1; (3) needs more testing if p = 1. (c) Comparison Test: Given

and

with

ak > 0, bk > 0 for all ks, and a1 ≤ b1, a2 ≤ b2,... ak ≤ bk for all ks: (1) If

converges, then

converges.

(Note that if the bigger series converges, then the smaller series converges.) (2) If

diverges, then

diverges.

(Note that if the smaller series diverges, then the bigger series diverges.) (d) Limit Comparison Test: Given

Given

and

with ak > 0, bk > 0 for all ks, and let

if 0 < p < ∞, then both series converge or both series diverge. (e) Integral Test: Given

, for all ks, and ak = f(k) for some function f(x),

if the function f is positive, continuous, and decreasing for all x ≥ 1, then ∞

and

33. Maclaurin Series:

, either both converge or both diverge.

34. Taylor Series:

Partial Sum:

35. Testing a Power Series for Convergence Given:

(1) Use Ratio Test to find values of x for absolute convergence. (2) Exactly one of the following cases will occur: (a) Series converges only at x = a. (b) Series converges absolutely for all x ∈ R. (c) Series converges on all x ∈ (a – R, a + R) and diverges for x < a – R or x > a + R. At the endpoints x = a – R and x = a + R, use an Integral Test, an Alternating Series Test, or a Comparison Test to test for convergence....