Calculus Study guides, Class notes & Summaries

power rule - ANSWER-if y = x^k -- then, k * x^k⁻¹ product rule - ANSWER-f(x) • g(x) then [f(x)•g(x)' + f(x)'•g(x)] quotient rule - ANSWER-D(x) • N'(x) - N(x) •D'(x) --------------------- [D(x)]² extended power rule - ANSWER-[g(x)]^k = k[g(x)]^k-1 • g(x)' chain rule - ANSWER-[f of g(x)]' = [f(g(x))]' = f'(g(x)) • g'(x) high order derivatives - ANSWER- what does a second derivative represent? - ANSWER-the rate of change of the rate of change ...

equation of a tangent line - ANSWER-y= f'(x)(x-x1)+f(x) difference quotient/ average rate of change (AROC) - ANSWER-f(x+h)-f(x)/h point slope form of a line - ANSWER-y-y1=m(x-x1) derivative is the same as... - ANSWER--slope of the tangent line -instantaneous rate of change -marginal (cost, profit, revenue) slope of the secant line - ANSWER-msec= (y2-y1)/(x2-x1) constant rule - ANSWER-if C is a real number, then the derivative of C =0 power rule - ANSWER-if n is a nonnegative...

Points of Inflection - ANSWER-If (c, f(c)) is a point of inflection of the graph of f, then either f"(c)=0 or f" is undefined at c Critical Value - ANSWER-Where first derivative is 0 or undefined Find absolute extrema - ANSWER-Use critical values and END POINTS in the function Horizontal Asymptote Rules - ANSWER-If m>n: NO HA If m=n: HA = co-eff of m/co-eff of n If m<n: HA: y = 0 Mean Value Theorem - ANSWER-if f(x) is continuous on [a,b] and differentiable on (a,b), there i...

definition of a limit - ANSWER-A limit is a number that a function approaches as x approaches a value Limit equation - ANSWER-f(x+h)-f(x)/h e ln(x) - ANSWER-x e^x e^y - ANSWER-e^(x+y) e^xy - ANSWER-(e^x)^y lne^x - ANSWER-x ln(xy) - ANSWER-ln(x)+ln(y) ln(x^y ) - ANSWER-ylnx tan theta - ANSWER-sin theta/cos theta secant of theta - ANSWER-1/cos theta cot theta - ANSWER-cos theta/sin theta csc theta - ANSWER-1/sin theta lim x->c (f(x) + g(x)) - ANSWER-lim x-&gt...

What will calculus enable you to do? - ANSWER-Find the volumes of oddly shaped objects Consider the function y=x^2-2x+1. What is the slope of the tangent line at x=2? - ANSWER-2 What is the average rate of change of the function y=2x^2+3 between x=x1 and x=x2 - ANSWER-2(x2+X1) What is the slope of the secant line of the function y=4x^2-2x+1 between x=3 and x=6 - ANSWER-34 A body is thrown upward so that its height h at the end of t seconds is h=160t-16t^2. What is its max height? - A...

Power Rule - ANSWER-d/dx x^n = nx^n-1 d/dx [c] - ANSWER-0 d/dx [sinx] - ANSWER-cosx d/dx [cosx] - ANSWER--sinx d/dx [e^x] - ANSWER-e^x Product Rule - ANSWER-d/dx [f(x) g(x)] = f'g+g'f Quotient Rule - ANSWER-d/dx [f(x)/g(x)] = f'g-g'f/g^2 d/dx [tanx] - ANSWER-sec^2x d/dx [secx] - ANSWER-secxtanx d/dx [cotx] - ANSWER--csc^2x d/dx [cscx] - ANSWER--cscxcotx Chain Rule - ANSWER-d/dx [f(g(x))] = f'(g(x)) * g'(x) Inside/Outside Properties - ANSWER-y=(inside)^n ...

volume equation - ANSWER-v=4/3 r^3 related rate problems - ANSWER-treat all variables as functions of time, use implicit differentiation to produce equalities which can be solved for the desired rates linear approximation equation - ANSWER-f'(a)(x-a)+f(a) linear approximation - ANSWER-approximating functions using the equation of a line, also known as tangent line approxiamtion equation for dy - ANSWER-f'(x)dx or f'(x)delta x dy - ANSWER-differential, dependent on both the valu...

INTG(sin(x) dx) - ANSWER--cos(x) + C INTG(cos(x) dx) - ANSWER-sin(x) + C INTG((sec(x))^2 dx) - ANSWER-tan(x) + C INTG((csc(x))^2 dx) - ANSWER--cot(x) + C INTG(sec(x)*tan(x) dx) - ANSWER-sec(x) + C INTG(csc(x)*cot(x) dx) - ANSWER--csc(x) + C INTG(b^x dx) - ANSWER-(1/ln(b))*b^x + C INTG(1/x dx) - ANSWER-ln(abs(x)) + C INTG((1/sqrt(1-x^2)) dx) - ANSWER-arcsin(x) + C INTG(-(1/sqrt(1-x^2)) dx) - ANSWER-arccos(x) + C INTG((1/(1+x^2)) dx) - ANSWER-arctan(x) + C Rational...

∫ 2x / (x² - 1) - ANSWER-Method: U sub HW 6 #1 Answer: ln(x+1) + ln(x-1) + C ∫ x - 8 x² + 2x - 8 - ANSWER-Method: Partial Fractions HW 6 #3 Answer: 2 ln(x+4) - ln(x-2) + C ∫ 9x - 25 3x² - 13x + 4 - ANSWER-Method: Partial Fractions HW 6 #5 Answer: 2 ln(3x-1) + ln(x-4) + C ∫ x² + 4x + 5 x³ + x² + x + 1 - ANSWER-Method: Partial Fractions HW 6 #7 Answer: 4 tan⁻¹(x) + ln(x+1) + C ∫ 4x² + 11 x³ + x² + 4x + 4 - ANSWER-Method: Partial Fractions HW 6 #9 An...

Vertical Asymptote - ANSWER-Set Denominator equal to zero Horizontal asymptote - ANSWER-set numerator equal to zero Rolles theorem states there is... - ANSWER-a tangent line at some point on a closed interval [a,b] if the following hold true a) continuous b) differentiable c) f(a)=f(b) Integration is a... - ANSWER-continuous addition process Intermediate Value Theorem: Suppose f is a function continuous at every point on the interval [a,b]: - ANSWER-a) f will take on every value b...