Limit Definition of the Derivative
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Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited)
Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited) Limits as ‘ x ’ Approaches Infinity At times you’ll need to know the behavior of a function or an expression as the inputs get increasingly larger … larger in the positive and negative directions. We can evaluate this using the limit limf ( x ) and limf ( x ) . x→ ∞ x→ −∞ Obviously, you cannot use direct substitution when it comes to these limits. Infinity is not a number, but a way of denoting how the inputs for a function can grow without any bound. You see limits for x approaching infinity used a lot with fractional functions. 1 Ex) Evaluate lim using a graph. x→ ∞ x A more general version of this limit which will help us out in the long run is this … GENERALIZATION For any expression (or function) in the form CONSTANT , this limit is always true POWER OF X CONSTANT lim = x→ ∞ xn HOW TO EVALUATE A LIMIT AT INFINITY FOR A RATIONAL FUNCTION Step 1: Take the highest power of x in the function’s denominator and divide each term of the fraction by this x power. Step 2: Apply the limit to each term in both numerator and denominator and remember: n limC / x = 0 and lim C= C where ‘C’ is a constant. x→ ∞ x→ ∞ Step 3: Carefully analyze the results to see if the answer is either a finite number or ‘ ∞ ’ or ‘ − ∞ ’ 6x − 3 Ex) Evaluate the limit lim . x→ ∞ 5+ 2 x 3− 2x − 5 x 2 Ex) Evaluate the limit lim . x→ ∞ 2x + 7 5x+ 2 x −2 Ex) Evaluate the limit lim . -
Section 8.8: Improper Integrals
Section 8.8: Improper Integrals One of the main applications of integrals is to compute the areas under curves, as you know. A geometric question. But there are some geometric questions which we do not yet know how to do by calculus, even though they appear to have the same form. Consider the curve y = 1=x2. We can ask, what is the area of the region under the curve and right of the line x = 1? We have no reason to believe this area is finite, but let's ask. Now no integral will compute this{we have to integrate over a bounded interval. Nonetheless, we don't want to throw up our hands. So note that b 2 b Z (1=x )dx = ( 1=x) 1 = 1 1=b: 1 − j − In other words, as b gets larger and larger, the area under the curve and above [1; b] gets larger and larger; but note that it gets closer and closer to 1. Thus, our intuition tells us that the area of the region we're interested in is exactly 1. More formally: lim 1 1=b = 1: b − !1 We can rewrite that as b 2 lim Z (1=x )dx: b !1 1 Indeed, in general, if we want to compute the area under y = f(x) and right of the line x = a, we are computing b lim Z f(x)dx: b !1 a ASK: Does this limit always exist? Give some situations where it does not exist. They'll give something that blows up. -
13 Limits and the Foundations of Calculus
13 Limits and the Foundations of Calculus We have· developed some of the basic theorems in calculus without reference to limits. However limits are very important in mathematics and cannot be ignored. They are crucial for topics such as infmite series, improper integrals, and multi variable calculus. In this last section we shall prove that our approach to calculus is equivalent to the usual approach via limits. (The going will be easier if you review the basic properties of limits from your standard calculus text, but we shall neither prove nor use the limit theorems.) Limits and Continuity Let {be a function defined on some open interval containing xo, except possibly at Xo itself, and let 1 be a real number. There are two defmitions of the· state ment lim{(x) = 1 x-+xo Condition 1 1. Given any number CI < l, there is an interval (al> b l ) containing Xo such that CI <{(x) ifal <x < b i and x ;6xo. 2. Given any number Cz > I, there is an interval (a2, b2) containing Xo such that Cz > [(x) ifa2 <x< b 2 and x :;Cxo. Condition 2 Given any positive number €, there is a positive number 0 such that If(x) -11 < € whenever Ix - x 0 I< 5 and x ;6 x o. Depending upon circumstances, one or the other of these conditions may be easier to use. The following theorem shows that they are interchangeable, so either one can be used as the defmition oflim {(x) = l. X--->Xo 180 LIMITS AND CONTINUITY 181 Theorem 1 For any given f. -
Lecture 9: Partial Derivatives
Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables. -
Differentiation Rules (Differential Calculus)
Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. -
Two Fundamental Theorems About the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b]. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
CHAPTER 3: Derivatives
CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems. (Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x. (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x. (By “arbitrary,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval []a, x , where a < x , is the line that passes through the points a, fa and x, fx. -
Infinitesimal Calculus
Infinitesimal Calculus Δy ΔxΔy and “cannot stand” Δx • Derivative of the sum/difference of two functions (x + Δx) ± (y + Δy) = (x + y) + Δx + Δy ∴ we have a change of Δx + Δy. • Derivative of the product of two functions (x + Δx)(y + Δy) = xy + Δxy + xΔy + ΔxΔy ∴ we have a change of Δxy + xΔy. • Derivative of the product of three functions (x + Δx)(y + Δy)(z + Δz) = xyz + Δxyz + xΔyz + xyΔz + xΔyΔz + ΔxΔyz + xΔyΔz + ΔxΔyΔ ∴ we have a change of Δxyz + xΔyz + xyΔz. • Derivative of the quotient of three functions x Let u = . Then by the product rule above, yu = x yields y uΔy + yΔu = Δx. Substituting for u its value, we have xΔy Δxy − xΔy + yΔu = Δx. Finding the value of Δu , we have y y2 • Derivative of a power function (and the “chain rule”) Let y = x m . ∴ y = x ⋅ x ⋅ x ⋅...⋅ x (m times). By a generalization of the product rule, Δy = (xm−1Δx)(x m−1Δx)(x m−1Δx)...⋅ (xm −1Δx) m times. ∴ we have Δy = mx m−1Δx. • Derivative of the logarithmic function Let y = xn , n being constant. Then log y = nlog x. Differentiating y = xn , we have dy dy dy y y dy = nxn−1dx, or n = = = , since xn−1 = . Again, whatever n−1 y dx x dx dx x x x the differentials of log x and log y are, we have d(log y) = n ⋅ d(log x), or d(log y) n = . Placing these values of n equal to each other, we obtain d(log x) dy d(log y) y dy = . -
Calculus Formulas and Theorems
Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. -
Antiderivatives 307
4100 AWL/Thomas_ch04p244-324 8/20/04 9:02 AM Page 307 4.8 Antiderivatives 307 4.8 Antiderivatives We have studied how to find the derivative of a function. However, many problems re- quire that we recover a function from its known derivative (from its known rate of change). For instance, we may know the velocity function of an object falling from an initial height and need to know its height at any time over some period. More generally, we want to find a function F from its derivative ƒ. If such a function F exists, it is called an anti- derivative of ƒ. Finding Antiderivatives DEFINITION Antiderivative A function F is an antiderivative of ƒ on an interval I if F¿sxd = ƒsxd for all x in I. The process of recovering a function F(x) from its derivative ƒ(x) is called antidiffer- entiation. We use capital letters such as F to represent an antiderivative of a function ƒ, G to represent an antiderivative of g, and so forth. EXAMPLE 1 Finding Antiderivatives Find an antiderivative for each of the following functions. (a) ƒsxd = 2x (b) gsxd = cos x (c) hsxd = 2x + cos x Solution (a) Fsxd = x2 (b) Gsxd = sin x (c) Hsxd = x2 + sin x Each answer can be checked by differentiating. The derivative of Fsxd = x2 is 2x. The derivative of Gsxd = sin x is cos x and the derivative of Hsxd = x2 + sin x is 2x + cos x. The function Fsxd = x2 is not the only function whose derivative is 2x. The function x2 + 1 has the same derivative. -
Numerical Differentiation
Jim Lambers MAT 460/560 Fall Semester 2009-10 Lecture 23 Notes These notes correspond to Section 4.1 in the text. Numerical Differentiation We now discuss the other fundamental problem from calculus that frequently arises in scientific applications, the problem of computing the derivative of a given function f(x). Finite Difference Approximations 0 Recall that the derivative of f(x) at a point x0, denoted f (x0), is defined by 0 f(x0 + h) − f(x0) f (x0) = lim : h!0 h 0 This definition suggests a method for approximating f (x0). If we choose h to be a small positive constant, then f(x + h) − f(x ) f 0(x ) ≈ 0 0 : 0 h This approximation is called the forward difference formula. 00 To estimate the accuracy of this approximation, we note that if f (x) exists on [x0; x0 + h], 0 00 2 then, by Taylor's Theorem, f(x0 +h) = f(x0)+f (x0)h+f ()h =2; where 2 [x0; x0 +h]: Solving 0 for f (x0), we obtain f(x + h) − f(x ) f 00() f 0(x ) = 0 0 + h; 0 h 2 so the error in the forward difference formula is O(h). We say that this formula is first-order accurate. 0 The forward-difference formula is called a finite difference approximation to f (x0), because it approximates f 0(x) using values of f(x) at points that have a small, but finite, distance between them, as opposed to the definition of the derivative, that takes a limit and therefore computes the derivative using an “infinitely small" value of h.