Calculus 1 - Final Study Guide

Math Final Information Note: Not all information on this sheet will be tested, but it does have every possible piece of information that can be tested. 1 General summary: Note: this is very general. Some topics have a lot of subtopics. • Limits • continuity • Derivatives • Implicit Differentiation • Maximum, minimums, critical points, inflection points • Optimization problems • Related rates • linear approximation • Newton’s method • Antiderivatives • Riemann sums • Definite integrals • indefinite integrals • Areas between curves • volumes of rotation. 2 2.1 L1 (The Tangent and Velocity Problems) • Know how to calculate the secant line between two points. 3 2.2+2.3 L2 (The Limit) Understand what a limit is. Know when limits exist/not exist – Left and Right limit must agree It doesn’t exist bc it oscillates between 1 & -1 – Know why lim  sin x1 does not exist. x→0 Know the limit laws (addition,division, multiplication, subtraction). [Direct substitution if it is continuous] x→0 4 2.5 L3 (Continuity) Understand the definition of continuity x f(a) is defined, lim as x approaches a of f(x) is defined, & lim as x approaches a f(x) = f(a) • Know when discontinuities occur, and the types of discontinuities. • Example: Where is x − 1 continuous? • 2.6 L3 (Limits at infinity and horizontal asymptotes) • Understand limits at infinity lim f (x). x→∞ • Know limits at infinity of common functions Example: lim x→∞ 1 xr = 0 for r > 0. • Know how to compute limits at infinity of rational functions. Example: lim x→∞ 5x2 +1 3x2 +x • Know when a function has horizontal/vertical asymptotes. 6 2.7 L4 (Rates of Change and Derivatives) • Know the definition of a derivative. “Limit of a secant line/difference quotient” • Be able to compute a derivative using the definition of a derivative (No power rule allowed). 7 2.8 L4+L5 (Derivative as a function) • Understand the derivative as a function. • Know when a function is differentiable/not differentiable (sharp corner, discontinuity) • Higher order derivatives: f ′′ (x) 3.1 L6 (Derivatives of polynomials and Exponential functions) 8 • Power rule • derivative rules (addition, subtraction, scalar multiplication) • Exponential derivative: • 9 d dx dx ax . ex = ex 3.2 L6 (Product and Quotient rules) • Product Rule • Quotient Rule 10 d d/dx f(x)g(x) = f’(x)g(x) + f(x)g’(x) d/dx f(x)/g(x) = g(x)f’(x) - f(x)g’(x) / g(x)^2 3.10 L7 (Linear approximation) • Understand how to do a linear approximation. • Understand what it means: “using the tangent line at a known value to estimate the function” • Understand if a linear approximation is an over/underestimate (Check concavity). 11 4.1 L8 (Maximum and Minimum Values) • Local and Absolute Min (How to find, definitions, and know what happens at endpoints) • Fermat’s theorem (Also know that a horizontal tangent does not always mean a local min or max. See f (x) = x3 ) • Critical points. • Extreme Value Theorem and Closed internal test. 12 4.3 L8 (How derivatives affect the shape of a curve) • Increasing/Decreasing definition and Increasing/Decreasing test • First Derivative test • Concavity definition, inflection point definition. How to determine concavity/inflection points. • Second derivative test 13 4.7 L9 (Optimization Problems) • Optimization Problems (This topic is very important and will likely be a word problem. Know how to set up and solve various different kinds of optimization problems) • First derivative test for absolute extreme values. 14 3.3 L10 (Trig derivatives) • Know the derivatives of trig functions, and be able to use them. • Know the following limit: lim x→0 to use this limit is important). sin(x) x = 1, and be able to use it to solve other trig limits (Knowing how 15 3.4 L11 (Chain Rule) • Know how to use the chain rule. • Know the derivative of the general exponential function ax . 16 = ln(a)* a^x 3.5 L12 (Implicit differentiation) • Know how to use implicit differentiation. • Know inverse trig derivatives. 17 3.6 L13 (Derivatives of logarithmic functions) • Know the derivative of the natural log and the general logarithm loga (x) • Know how to use logarithmic differentiation (It is also important for you to recognize, when to use logarithmic differentiation for example a complicated function 2 sin(x)x ) 18 √ x2 +6 sin(x)ex or a function to a function power . 3.9 L14 (Related Rates) • Know how to set up and solve various related rates problem. • Know how to find a relationship between two rates of change. This is done through implicit differentiation. 19 4.4 L15 (Indeterminate forms and L’Hopital’s Rule) • Know how to recognize the various indeterminate forms. • Know how to apply L’Hopital’s Rule to find a limit. • Remember that you cannot apply L’Hopital directly to an “addition”, “multiplication”, or “exponential” indeterminate form. You must transform these forms to look like division. 20 4.2 L16 (Mean Value Theorem) • You should know the statements and meaning of the Mean Value theorem and Rolle’s theorem. • You have to be able to apply it to a proof. • “Show there is exactly one real root”, “If f ′ (x) ≤ 5, what is the highest possible value for f ? in some interval”. “Show that there is a point in the interval (a, b), where the tangent slope is equal to the secant slope of the end points” 21 4.8 L16 (Newton’s Method) • Recognize when to use Newton’s method (Used to find zeros): “estimate a root of x3 + x − 1” or “estimate when x = cos(x)” • Know the formula to generate the next term in Newton’s method and know how to use it: xn+1 = xn − 22 f (xn ) f ′ (xn ) 4.9 L17 (Antiderivatives) • Know what an antiderivative is. the opposite of a derivative, it reverses what the derivative does • Know that you can add/subtract/scalar-multiply antiderivatives. • Know that there is more than one antiderivaitve and you need +C in the most general antiderivative • Know how to use the antiderivative in F T C 2. 23 L18 (Area and Distance Problem) • Know how to estimate the area under a curve using rectangles (Riemann sum) with right, left, middle sample points. • Know how to calculate a Riemann sum with a finite number of rectangles (Not a limit). Example: estimate the under x2 from 0 to 1 using a right Riemann sum with 6 rectangles. 24 5.2 L18+L19 (Definite Integral) • Know the Definition of a definite integral (Limit of a Riemann sum). Also, know that it is equal to the area under a curve • Know how to calculate a definite integral using the “limit of a Riemann sum” or “definition of a Riemann integral” by simplifying a summation and taking the limit (NOTE: for this kind of problem you cannot use FTC). n • Know the values: P i=1 n i, P i=1 i2 , and n P i=1 i3 . • Know how to calculate an integral using geometry (directly calculate the area under a curve using geometry) • Know how to use integral properties/laws to simplify integrals (addition, scalar multiplication, splitting the interval, switching the limits) 25 5.3 L20 (Fundamental theorem of calculus) • Know what this function means g(x) = Ra xf (t)dt. • Be able to use FTC 1 to calculate the derivative of functions that look like g(x). Also, remember that you might need to split the interval and the chain rule (Example: g(x) = is not a constant and the upper limit is not just x). xR xf2(t)dt. The lower limit • Be able to use FTC 2 to calculate a definite integral. 26 5.5 L21 (Substitution Rule) • Know what an indefinite integral is. (Antiderivative need +C in your answer) • Know how to use u-substitution in calculating indefinite integrals. • Know how use u-substitution for definite integrals (need to change limits of integration). 27 6.1 L22 (Areas between curves) • Know how to compute areas between curves. • Know when to use two integrals (curves switch places) • Know when/how to integrate over y . 28 6.2 L23 (Volumes) • Know how to compute the volumes of solids when you rotate an area around the x or y axis. • You have to take the Riemann sum of cylinders/rings. or • you integrate over flat discs/ flat rings. 29 6.3 L24 (Volumes/shell method) • Know how to compute the volumes of solids like in 6.2 except with cylindrical shells instead. 30 Important Things to Know (Memorize) Derivatives/antiderivatives: NOTE: Be able to go backwards and remember the antiderivative and do not forget the +C for antiderivaitves Remember this new derivative: d dx 1 x ln |x| = for all x = 0 (Remember the absolute value). Trig Derivatives Inverse Trig Derivatives • d dx • d sin(x) = cos(x) dx cos(x) = − sin(x) • d dx tan(x) = sec2 (x) • d dx cot(x) = − csc2 (x) • d dx sec(x) = sec(x) tan(x) • d dx csc(x) = − csc(x) cot(x) • d dx arcsin = • d dx arccos = − √ • d dx arctan = • d dx arccot = − • d dx arcsec = • d dx arccsc = − Power and Exponential Derivatives 1 √ 1−x2 1 1−x2 1 1+x2 1 1+x2 . 1 √ x x2 −1 1 √ x x2 −1 Log Derivatives • d dx • d dx • d dx ex = ex • x a = ln(a)a x • a x = ax a−1 d 1 x dx ln |x| = d dx logb (x) = 1 ln(b)x Summation Identities: • • • • • • n P i=1 c=c·n n n P i=1 cai = c P i=1 a i + bi = n P i=1 n P i=1 n P i=1 n n n P i=1 ai P i=1 ai + n a i + bi = i= P i=1 P i=1 bi n b ai − i=1 P i n(n+1) 2 i2 = n(n+1)(2n+1) 6 n 2  i3 = n(n+1) 2 Integral Properties/Laws: • 1. P i=1 Ra bcdx = c(b − a), where c is a constant. 2. Rb R ba g(x)dx. Rb [f (x) + g(x)]dx = a f (x)dx + a 3. R ba f (x)dx, where c is a constant. Rb cf (x)dx = c a 4. Rb R ba g(x)dx. Rb [f (x) − g(x)]dx = a f (x)dx − a 5. R bc f (x)dx, where c is between a and b. Rc Rb f (x)dx = a f (x)dx + a b 6. If f (x) ≥ 0 for a ≤ x ≤ b, then R f (x)dx ≥ 0. a b b 7. If f (x) ≥ g(x) for a ≤ x ≤ b, then R f (x)dx ≥ R g(x)dx. a a 8. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then b m(b − a) ≤ Z f (x)dx ≤ M (b − a) a 9. Ra R ba f (x)dx f (x)dx = − b Theorems: You must be able to apply these theorems. Know when and why they are used. Squeeze Theorem: If f (x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and lim f (x) = lim h(x) = L x→a x→a then limg(x) = L x→a Intermediate Value Theorem: If f is continuous on closed interval [a, b] and d is a number between f (a) and f (b) (f (a) ≤ N d ≤ f (b) or f (b) ≤ d ≤ f (a)), then we can find number c in (a, b) such that f (c) = d. Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (x) and an absolute minimum value f (d) at some numbers c and d in [a, b]. Rolle’s Theorem: Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). 3. f (a) = f (b) Then there is a number c in (a, b) such that f ′ (c) = 0. Mean Value Theorem: Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval; [a, b]. 2. f is differentiable on the open interval (a, b). then there is a number c in (a, b) such that f ′ (c) = f (b) − f (a) b−a Theorems Continued: The Fundamental Theorem of Calculus, part 1: If f is continuous on [a, b], then we have function g defined by g(x) = Z x f (t)dt, a ≤ x ≤ b a where g is continuous on [a, b] and differentiable on (a, b), where g ′ (x) = f (x) The Fundamental Theorem of Calculus, part 2: If f is continuous on [a, b], then Z b a where F is any antiderivative of f . f (x)dx = F (b) − F (a) Useful identities and formulas: trig identities: • sin2 (x) + cos2 (x) = 1. • tan2 (x) + 1 = sec2 (x) • cos2 (x) = 1+cos(2x) 2 Also know values of trig functions (Ex: cos(6 )π =?), and when they are zero, undefined, negative, and positive. Algebra rules: • ln(ab) = ln(a) + ln(b) • ln(a/b) = ln(a) − ln(b) • ln(ab ) = b ln(a) • x = eln(x) • x = ln(ex ) • ea+b = ea · eb • eab = (ea )b Note: (ea )b is not the same as ea b Note: The above are true for other bases, not just e • x = sin(arcsin(x)) (This is true for other trig functions) Geometry: NOTE: Geometry will be used in related rates, optimization, calculating a definite integral, and volumes of rotation. Note: Do not just memorize these. Sometimes we have to modify these equations: Like half a circle, half a sphere, rectangle without a side. Pythagorean Theorem: z 2 = x2 +y 2 , where z is the length of the hypotenuse (Largest side) of a right triangle and x, y are the other side lengths. p Distance between two points (a, b) and (x, y): d = (a − x)2 + (b − y)2 For the following, r is the radius, h is the height. Circle Area: A = πr2 . Circle Circumference: C = 2πr . Sphere Volume: V = 43 πr 3 . Cone Volume: 1 πr 2 h. 3 Cylinder Volume: V = πr 2 h. Cylinder surface area: S = 2πr 2 + 2hπr. Know rectangle area and perimeter. A = bh and P = 2b + 2h Know rectangular prism volume and surface area. V = bhw S = 2bw + (2b + 2w)h