Trigonometric, Differentiation and Integration Cheat Sheet, Cheat Sheet of Calculus

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Trigonometric formulas

Differentiation formulas

Integration formulas

y = D + A sin B x ( − C ) A is amplitude B is the affect on the period (stretch or shrink)

C is vertical shift (left/right) and D is horizontal shift (up/down)

Limits:

0 0

lim sin^1 lim sin^0 lim 1 cos 0

x x x

x x x

−> x −>∞ x −> x

6. Let f be differentiable for a < x < b and continuous for a ≤ x ≤ b****. a. If f'(x) > 0 for every x in (a,b) , then f is increasing on [a,b]. b. If f'(x) < 0 for every x in (a,b) , then f is decreasing on [a,b]. 7. Suppose that f''(x) exists on the interval (a,b). a. If f''(x) > 0 in (a,b) , then f is concave upward in (a,b). b. If f''(x) < 0 in (a,b) , then f is concave downward in (a,b).

To locate the points of inflection of y = f(x) , find the points where f''(x) = 0 or where f''(x) fails to exist. These are the only candidates where f(x) may have a point of inflection. Then test these points to make sure that f''(x) < 0 on one side and f''(x) > 0 on the other.

8. Mean value theorem

If f is continuous on [a,b] and differentiable on (a,b), then there is at least one number c

in (a,b) such that.

9. Continuity

If a function is differentiable at a point x = a , it is continuous at that point. The converse is false, i.e. continuity does not imply differentiability.

10. L'Hôpital's rule

If is of the form or , and if exists, then.

11. Area between curves

If f and g are continuous functions such that f(x) ≥ g(x) on [a,b] , then the area between the curves is.

12. Inverse functions a. If f and g are two functions such that f(g(x)) = x for every x in the domain of g , and, g(f(x)) = x , for every x in the domain of f , then, f and g are inverse functions of each other. b. A function f has an inverse if and only if no horizontal line intersects its graph more than once. c. If f is either increasing or decreasing in an interval, then f has an inverse. d. If f is differentiable at every point on an interval I , and f'(x) ≠ 0 on I , then g = f-1(x) is differentiable at every point of the interior of the interval f(I) and

.

13. Properties of y = ex a. The exponential function y = ex^ is the inverse function of y = ln x. b. The domain is the set of all real numbers, − ∞ < x < ∞. c. The range is the set of all positive numbers, y > 0.

d. e.

14. Properties of y = ln x a. The domain of y = ln x is the set of all positive numbers, x > 0. b. The range of y = ln x is the set of all real numbers, − ∞ < y < ∞. c. y = ln x is continuous and increasing everywhere on its domain. d. ln (ab) = ln a + ln b. e. ln (a / b) = ln a − ln b. f. ln ar^ = r ln a. 15. Fundamental theorem of calculus

, where F'(x) = f(x), or.

16. Volumes of solids of revolution a. Let f be nonnegative and continuous on [a,b] , and let R be the region bounded above by y = f(x) , below by the x-axis, and the sides by the lines x = a and x = b. b. When this region R is revolved about the x-axis, it generates a solid (having circular cross sections) whose volume. c. When R is revolved about the y-axis, it generates a solid whose volume . 17. Particles moving along a line a. If a particle moving along a straight line has a positive function x(t) , then its instantaneous velocity v(t) = x'(t) and its acceleration a(t) = v'(t). b. v(t) = ∫ a(t)dt and x(t) = ∫ v(t)dt. 18. Average y-value

The average value of f(x) on [a,b] is.