Definitions, Formulas, Tests, and Theorems to Memorize 1) lim f ( x ) xa exists iff lim f ( x) lim f ( x) x a x a 2) f(x) is continuous at x=a iff f (a) lim f ( x) x a 3) Intermediate Value Theorem: if f(x) is continuous on [a,b], and k is between f(a) and f(b), f(c) = k for some c in [a,b]. 4) Mean Value Theorem: if f(x) is continuous on [a,b] and differentiable on (a,b), ∃ a c in (a,b) s.t. f '(c) 5) Definition of the Derivative: lim h 0 f ( x h) f ( x ) h 6) Average Rate of Change: of f(x) over the interval [a,b] is 7) Average Value: of f(x) over the interval [a,b] is b a lim or xa f (b) f (a ) ba f ( x) f (a) xa f (b) f ( a ) ba f ( x)dx ba 8) If (a,b) is on the graph of f(x) and 𝑔(𝑥) = 𝑓 −1 (𝑥) then g '(b) 1 f '( a ) 9) If f’’(x)>0, f(x) is concave up and f’(x) is increasing; if f’’(x)<0, f(x) is concave down and f’(x) is decreasing 10) First Derivative Test: if f’(x) changes from positive to negative at x=c, then f(x) has a local maximum at c 11) First Derivative Test: if f’(x) changes from negative to positive at x=c, then f(x) has a local minimum at c 12) Second Derivative Test: if f’(c)=0 and f’’(c)<0 then f(x) has a local maximum at c 13) Second Derivative Test: if f’(c)=0 and f’’(c)>0 then f(x) has a local minimum at c 14) f(x) has a point of inflection when f’’(x) changes sign 15) Tangent lines and Euler’s Method’s steps are below the curve and are thus an under approximation when f’’(x)>0 16) Tangent lines and Euler’s Method’s steps are above the curve and are thus an over approximation when f’’(x)<0 17) Secant lines are above the curve and thus Trapezoidal Approximations are over approximations when f’’(x)>0 18) Secant lines are below the curve and thus Trapezoidal Approximations are under approximations when f’’(x)<0 19) Area of a trapezoid: 1 h b1 b2 2 20) Fundamental Theorem of Calculus: for any constant a, d x f (t )dt = f ( x ) dx a 21) FTC Extension: if the bounds are differentiable functions, then d h( x) f (t )dt = f h( x) h '( x) f g ( x) g '( x) dx g ( x ) 22) Euler’s Method: the change in y can be approximated by dy x dx 23) Logistic Growth: the carrying capacity C = lim P (t ) occurs when t dP 0 dt 1 2 24) Logistic Growth: if the carrying capacity is C, then P(t) has an inflection point and grows fastest when P(t)= C 25) Arc Length (rectangular): of f(x) over the interval [a,b] is 26) L’Hôpital’s Rule: if lim x a b a 1 f '( x) dx 2 f ( x) f ( x) f '( x) 0 is of the form = lim , if it exists or then lim x a x a 0 g ( x) g ( x) g '( x) 27) Nth Term Test: for a n0 n , if lim an 0 , then the series diverges; if lim an 0 then the test is inconclusive n 28) P-series Test: the series 1 n n0 p n converges when p 1 and diverges when p 1 29) Geometric Series Test: the series k r n converges when r 1 and diverges when r 1 n 0 30) Ratio Test: examine lim n an1 L ; the series converges when L 1 and diverges when L 1 an 31) Direct Comparison Test: compare the series to one that is larger and converges or that is smaller and diverges 32) Slope of Tangent Line (Parametric): dy dy / dt dx dx / dt dy d 2 y derivative of dx 33) Parametric 2 Derivative: dx / dt dx 2 nd 34) Speed (Parametric) : given position vector x(t ), y(t ) , speed = x '(t ) y '(t ) 2 35) Total Distance = Arc Length (Parametric): for x(t ), y(t ) over [a,b]= 36) Slope of Tangent Line (Polar): 2 x '(t ) y '(t ) dt b 2 2 a dy dy / d , so first you must write y = r sin and x = r cos dx dx / d 37) Area between a polar curve r(θ) and the origin for is 1 2 r d 2