Math 1013 Applied Calculus I Student Notebook PDF

Student Notebook for MATH 1013: Applied Calculus I 1  Functions 1.1 A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. agroupof numbers 2 Interval Notat...

Student Notebook for MATH 1013: Applied Calculus I 1  Functions 1.1 A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. agroupof numbers 2 Interval Notation Is used to describe ranges of numbers in a convenient way. Words about ranges of numbers Notation you probably use Interval notation Values between a and b not including a or b a 1 x!1 p 10ex 21 5 x < f (x) < p made 2ex x 1 e 2 2 um fen him 5am 61 2e m a n n 0T 5 2 him fae 4 5 0 him f n 5 57 Derivatives and Rates of Change 2.7 The tangent line l to the curve y = f (x) at the point P (a, f (a)) is the line through the point P with slope f (x) f (a) rise m = lim x!a x a run provided the limit exists. acrofa The tangent line is the limit of the fca secant line between Panda e be 58 If we let x = a + h, we can rewrite the previous limit into HEIR h 0 m Fath f a Liz h h is the distance betweenthe two se valuesgeneratingthe secant line I th 59  Slopes of Tangent Lines, Derivatives, Velocities, and Instantaneous Rates of Change All the same thing derivative pascals slope rety triangle point this for Find an equation of the tangent line to the curve Taffy 3 1 s y = xBR 2x3 at (1, 1) iÉ u crease 242hm 2cm 36h 3m th 2 2 3 2mm bath 6am 2 my n him n gig n him 2x th 6m ban 2m m 2x bx slope at 1 1 h 0 slope m 2 1 66172 m 4 was 9 mar a 76 whenever 4 x 1 G 9 9 4 3 60  If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height after t seconds is given by H(t) = 10t 1.86t2 a) Find the velocity of the rock after one second. b) Find the velocity of the rock when t = a. c) When will the rock hit the surface? d) With what velocity will the rock hit the surface? a v th 1 864th 1061 1.86 1 2 m lotion 1 86 H2nth 10 1.86 10h 3.72h 1.86m ym h fig h 261 h 10 3.72 1.86h 6.28mg b a m ath 1.80 ath 10 a 1.8692 M 10 3.72A If ym h C H t 10 1.808 0 t 0 10 1804 0 t 95.4s d m 10 3.72 18 10ms 61  derivative If For the function g whose graph is given, arrange 0, g 0( 2), g 0(0), g 0(4) in increasing order g O 2029 4 49 C 2 62  f Dafoe x̅ a y L ten The Derivative as a Function 2.8 The derivative function is defined as f (x + h) f (x) f 0(x) = lim h!0 h A function is di↵erentiable at a if f 0(a) exists. It is di↵erentiable on an open interval (a, b), possibly infinite on either end, if it is di↵erentiable at every number on the interval. If f is di↵erentiable at a, then f is continuous at a Differentiable continuous fath Fca vs him forth Fcm plimo Tantanumber wantsatunction 63 Continuous Does Not Imply Di↵erentiable Consider the |x|. Where does it fail to be di↵erentiable? 10 101 n L 4 1 is f Ese few 1ft that In Left Right no limit no derivative 64 Given the following two graphs, where do the functions fail to be di↵erentiable? at x 4 at a 2 2 0 65  Higher Derivatives If f has a derivative function f 0, then f 0 may also have its own derivative. The notation is as follows 00 d2 f d2 y 9 d f (x), 2 , 2 dx dx dm If a function has an nth derivative, we define it as n n bracket derivative d f d y no bracket power f (n)(x), , dxn dxn f position f velocity howis thepositionchanging f acceleration now is thevelocitychanging f jerk how is the accelerationchanging 66 The figure shows the graphs of four functions. One is the position function of a car, one is the velocity, one is its acceleration, and one is its jerk. Identify each curve. position jerk acceleration velocity Andrew emailed the correct answer 67  Derivatives of Polynomials and Exponential Functions 3.1 If c is a constant, f (x), g(x) are di↵erentiable functions and n is a real number d (c) = 0 the Constant Rule dx d n (x ) = nxn 1 the Power Rule dx d d [cf (x)] = c f (x) the Constant Multiple Rule dx dx d d d [f (x) ± g(x)] = f (x) ± g(x) the Sum/Di↵erence Rule dx dx dx d x d x e = ex and b = bx ln(b) the Exponential Rule dx dx definition of the fath fin derivative him h 68 Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2. Derivative y 28 3 15m slopeofthetangent 2e 3 15m 72 Foranyvalue of x 2e 15m 0 2e 15m 37372 69  The Product and Quotient Rules 3.2 If f and g are di↵erentiable functions, then [f (x)g(x)]0 = f 0(x)g(x) + f (x)g 0(x) the Product Rule  0 f (x) f 0(x)g(x) f (x)g 0(x) = the Quotient Rule g(x) g(x)2 If f (x) = exg(x), where g(0) = 2 and g 0(0) = 5, find f 0(0). F x Effect f a e x e g g a f o e g O e g o 2 5 I 70 Suppose that f (5) = 1, f 0(5) = 6, g(5) = 3, and g 0(5) = 2. Find ⇣ ⌘0 ⇣ ⌘0 a) fg (5) b) fg (5) f f 9 a 9 m fire gca f a x 94 n f a g a 2 24 20 61 3 1 12 2,0 3 71 Derivatives of Trigonometric Functions 3.3 72 An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle ✓ with the plane, then the magnitude of the force is µW F = µ sin(✓) + cos(✓) where µ is a constant called the coefficient of friction. Find the rate of change of F with respect to ✓, and find when the rate of change is equal to 0. 0 MSino cos 6 MW M WSO since F 6 Msino cos012 m WCoso MWsino m sin 2µsinocos 056 M W cos MWSino tano are tan M 6 µ tano M MWCOSO Therate ofchange of dihde the forceis zero when are tan M byMWCoso 73  The Chain Rule 3.4 Outside In If g is di↵erentiable at x and f is di↵erentiable at g(x), then the composite function f (g(x)) is di↵eren- tiable at x and given by the product [f (g(x))]0 = f 0(g(x))g 0(x) In Leibniz notation, if y = f (u) and u = g(x) are both di↵erentiable functions, then dy dy du = dx du dx Find the derivative of simplify 1 a) (2x3 5x2 + 4)5 in exams b) p 3 2 x 1 m 1 y 5 2m 5m 4 bx tone Ei m 155 g 30m 502 2m 56 4 ET v21 2m 2 4215 3 74  Find the derivative of y = esin(2x) + sin(e2x). y es cos 2n 2 cos e e 2 2eˢ 2e cos e 75  If f and g are the functions whose graphs are shown, let u(x) = f (g(x)), v(x) = g(f (x)), and w(x) = g(g(x)). Find a) u0(1) b) v 0(1) c) w0(1) U a f g a W x g x g g a g a 1 f 9 i g i w i i g g i g 1 f 4 1 g 4 1 1 1 I x g f x f x 1 g f D f i g 2 2 1 2 2 76  Explicitfunctions vs Implicitfunctions lonely y fee a y sincy 0 Implicit Di↵erentiation 3.5 y x 2792 1 a Sinta The rules for di↵erentiation up until now have all been inf terms of explicit functions, functions of the form y = f (x). An implicit function is an equation of the form f (x, y) = 0. Many phenomena are modelled by implicit functions, and we often want to find the rates of changes in these models. Steps for finding an implicit derivative: 1. Start by shifting all terms in the equation to one side of the equals sign, to get f (x, y) = 0. 2. Take the derivative of each term in the equation. When taking the derivative of any term that has dy a y in it, use the chain rule and multiply the derivative by dx. dy 3. Solve for dx. 77  dy Find dx if x3 x2 + y 3 = 1. ³ m x 93- 1 I - a 39 2x 3 m² 2 0 y 78  Find the equation of the tangent line to the curve yesin(x) = x cos(y) at (0,0). yes COS y 0 Ix esin esin t y cos a i cos y re f sin y 0 e 0 1 0 0 1 0 y a a tb a b tangentformula y 1 x 0 0 y R 79 Derivatives of Logarithmic and Inverse Trigonometric Func- tions 3.6 d 1 d 1 ln(x) = , logb(x) = dx x dx x ln(b) d g 0(x) ln(g(x)) = dx g(x) Logarithmic Di↵erentiation - useful for calculating the derivatives of complicated functions involving products, quotients, or powers. Steps: 1. Take the natural logarithm of both sides of an equaton y = f (x) and use the Laws of Logarithms to expand the expression. 2. Di↵erentiate implicitly with respect to x. 3. Solve the resulting equation for y 0 and replace y by f (x). 80  take Ln of bothsides 0 e x cos2 (x) Find y if y = x2 +x+1. in in é LnCnt In a in 1 do not do this in y In é incos a n n att It's wrong In y x 2in cos n n re a 1 CI 2 y G 2tank 2 e cos a 1 stance x2 at 81  Derivatives of Inverse Trigonometric Functions memorize fish tests Find the derivative of 2) f (z) = earcsin(z f z earesina 22 F 82