Formulas

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: sin x =1 x −> 0 x lim sin x =0 x −>∞ x lim 1 − cos x =0 x −> 0 x lim Exponential Growth and Decay y = Ce kt Rate of Change of a variable y is proportional to the value of y dy = ky or dx y ' = ky Formulas and theorems 1. A function y=f(x) is continuous at x=a if i. f(a) exists ii. exists, and iii. 2. Even and odd functions 1. A function y = f(x) is even if f(-x) = f(x) for every x in the function's domain. Every even function is symmetric about the y-axis. 2. A function y = f(x) is odd if f(-x) = −f(x) for every x in the function's domain. Every odd function is symmetric about the origin. 3. Horizontal and vertical asymptotes 1. A line y = b is a horizontal asymptote of the graph of y = f(x) if either or . 2. A line x =a is a vertical asymptote of the graph of y =f(x) if either or . 4. Definition of a derivative 5. To find the maximum and minimum values of a function y = f(x), locate 1. . the points where f'(x) is zero or where f'(x) fails to exist 2. the end points, if any, on the domain of f(x). Note: These are the only candidates for the value of x where f(x) may have a maximum or a minimum 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 11. is of the form or , and if exists, then . 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. 14. 15. d. e. 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. 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. 18. . 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. Average y-value The average value of f(x) on [a,b] is .