Second Derivative Math165: Business Calculus

Total Page:16

File Type:pdf, Size:1020Kb

Second Derivative Math165: Business Calculus Second Derivative Math165: Business Calculus Roy M. Lowman Spring 2010, Week6 Lec2 Roy M. Lowman Second Derivative second derivative definition of f0 Definition (first derivative) df f(x + ∆x) − f(x) = f0 = lim (1) dx ∆x−>0 ∆x ∆f = lim (2) ∆x−>0 ∆x ∆f ≈ average slope of f(x) over delta x, f0 (3) ∆x avg (4) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f00 Definition (second derivative) d2f d f0(x + ∆x) − f(x)0 = f0(x) = lim (5) dx2 dx ∆x−>0 ∆x ∆f0 = lim (6) ∆x−>0 ∆x ∆f0 ≈ average slope of f0over∆x; f00 (7) ∆x avg (8) Roy M. Lowman Second Derivative second derivative definition of f00 f00(x) gives the concavity of function f(x) f00(x) is the rate of change of slope w.r.t x 00 f (x) is (+) where f(x) is concave up. (holds H2O) f00(x) is (−) where f(x) is concave down. (makes letter A) f00(x) can be used to find intervals where f(x) is concave up or concave down f00(x) = 0 or undefined where the concavity can change f00(x) = 0 at inflection points, but must check if actually IP. The Second Derivative Test can be used to determine what type of critical points where f0(x) = 0. f00(x) can tell you a lot about a function f(x) 00 ∆f0 00 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f00 f00(x) gives the concavity of function f(x) f00(x) is the rate of change of slope w.r.t x 00 f (x) is (+) where f(x) is concave up. (holds H2O) f00(x) is (−) where f(x) is concave down. (makes letter A) f00(x) can be used to find intervals where f(x) is concave up or concave down f00(x) = 0 or undefined where the concavity can change f00(x) = 0 at inflection points, but must check if actually IP.
Recommended publications
  • The Mean Value Theorem Math 120 Calculus I Fall 2015
    The Mean Value Theorem Math 120 Calculus I Fall 2015 The central theorem to much of differential calculus is the Mean Value Theorem, which we'll abbreviate MVT. It is the theoretical tool used to study the first and second derivatives. There is a nice logical sequence of connections here. It starts with the Extreme Value Theorem (EVT) that we looked at earlier when we studied the concept of continuity. It says that any function that is continuous on a closed interval takes on a maximum and a minimum value. A technical lemma. We begin our study with a technical lemma that allows us to relate 0 the derivative of a function at a point to values of the function nearby. Specifically, if f (x0) is positive, then for x nearby but smaller than x0 the values f(x) will be less than f(x0), but for x nearby but larger than x0, the values of f(x) will be larger than f(x0). This says something like f is an increasing function near x0, but not quite. An analogous statement 0 holds when f (x0) is negative. Proof. The proof of this lemma involves the definition of derivative and the definition of limits, but none of the proofs for the rest of the theorems here require that depth. 0 Suppose that f (x0) = p, some positive number. That means that f(x) − f(x ) lim 0 = p: x!x0 x − x0 f(x) − f(x0) So you can make arbitrarily close to p by taking x sufficiently close to x0.
    [Show full text]
  • AP Calculus AB Topic List 1. Limits Algebraically 2. Limits Graphically 3
    AP Calculus AB Topic List 1. Limits algebraically 2. Limits graphically 3. Limits at infinity 4. Asymptotes 5. Continuity 6. Intermediate value theorem 7. Differentiability 8. Limit definition of a derivative 9. Average rate of change (approximate slope) 10. Tangent lines 11. Derivatives rules and special functions 12. Chain Rule 13. Application of chain rule 14. Derivatives of generic functions using chain rule 15. Implicit differentiation 16. Related rates 17. Derivatives of inverses 18. Logarithmic differentiation 19. Determine function behavior (increasing, decreasing, concavity) given a function 20. Determine function behavior (increasing, decreasing, concavity) given a derivative graph 21. Interpret first and second derivative values in a table 22. Determining if tangent line approximations are over or under estimates 23. Finding critical points and determining if they are relative maximum, relative minimum, or neither 24. Second derivative test for relative maximum or minimum 25. Finding inflection points 26. Finding and justifying critical points from a derivative graph 27. Absolute maximum and minimum 28. Application of maximum and minimum 29. Motion derivatives 30. Vertical motion 31. Mean value theorem 32. Approximating area with rectangles and trapezoids given a function 33. Approximating area with rectangles and trapezoids given a table of values 34. Determining if area approximations are over or under estimates 35. Finding definite integrals graphically 36. Finding definite integrals using given integral values 37. Indefinite integrals with power rule or special derivatives 38. Integration with u-substitution 39. Evaluating definite integrals 40. Definite integrals with u-substitution 41. Solving initial value problems (separable differential equations) 42. Creating a slope field 43.
    [Show full text]
  • Chapter 9 Optimization: One Choice Variable
    RS - Ch 9 - Optimization: One Variable Chapter 9 Optimization: One Choice Variable 1 Léon Walras (1834-1910) Vilfredo Federico D. Pareto (1848–1923) 9.1 Optimum Values and Extreme Values • Goal vs. non-goal equilibrium • In the optimization process, we need to identify the objective function to optimize. • In the objective function the dependent variable represents the object of maximization or minimization Example: - Define profit function: = PQ − C(Q) - Objective: Maximize - Tool: Q 2 1 RS - Ch 9 - Optimization: One Variable 9.2 Relative Maximum and Minimum: First- Derivative Test Critical Value The critical value of x is the value x0 if f ′(x0)= 0. • A stationary value of y is f(x0). • A stationary point is the point with coordinates x0 and f(x0). • A stationary point is coordinate of the extremum. • Theorem (Weierstrass) Let f : S→R be a real-valued function defined on a compact (bounded and closed) set S ∈ Rn. If f is continuous on S, then f attains its maximum and minimum values on S. That is, there exists a point c1 and c2 such that f (c1) ≤ f (x) ≤ f (c2) ∀x ∈ S. 3 9.2 First-derivative test •The first-order condition (f.o.c.) or necessary condition for extrema is that f '(x*) = 0 and the value of f(x*) is: • A relative minimum if f '(x*) changes its sign y from negative to positive from the B immediate left of x0 to its immediate right. f '(x*)=0 (first derivative test of min.) x x* y • A relative maximum if the derivative f '(x) A f '(x*) = 0 changes its sign from positive to negative from the immediate left of the point x* to its immediate right.
    [Show full text]
  • Critical Points and Classifying Local Maxima and Minima Don Byrd, Rev
    Notes on Textbook Section 4.1 Critical Points and Classifying Local Maxima and Minima Don Byrd, rev. 25 Oct. 2011 To find and classify critical points of a function f (x) First steps: 1. Take the derivative f ’(x) . 2. Find the critical points by setting f ’ equal to 0, and solving for x . To finish the job, use either the first derivative test or the second derivative test. Via First Derivative Test 3. Determine if f ’ is positive (so f is increasing) or negative (so f is decreasing) on both sides of each critical point. • Note that since all that matters is the sign, you can check any value on the side you want; so use values that make it easy! • Optional, but helpful in more complex situations: draw a sign diagram. For example, say we have critical points at 11/8 and 22/7. It’s usually easier to evaluate functions at integer values than non-integers, and it’s especially easy at 0. So, for a value to the left of 11/8, choose 0; for a value to the right of 11/8 and the left of 22/7, choose 2; and for a value to the right of 22/7, choose 4. Let’s say f ’(0) = –1; f ’(2) = 5/9; and f ’(4) = 5. Then we could draw this sign diagram: Value of x 11 22 8 7 Sign of f ’(x) negative positive positive ! ! 4. Apply the first derivative test (textbook, top of p. 172, restated in terms of the derivative): If f ’ changes from negative to positive: f has a local minimum at the critical point If f ’ changes from positive to negative: f has a local maximum at the critical point If f ’ is positive on both sides or negative on both sides: f has neither one For the above example, we could tell from the sign diagram that 11/8 is a local minimum of f (x), while 22/7 is neither a local minimum nor a local maximum.
    [Show full text]
  • Multivariable and Vector Calculus
    Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space.
    [Show full text]
  • Concavity and Points of Inflection We Now Know How to Determine Where a Function Is Increasing Or Decreasing
    Chapter 4 | Applications of Derivatives 401 4.17 3 Use the first derivative test to find all local extrema for f (x) = x − 1. Concavity and Points of Inflection We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function. Figure 4.34(a) shows a function f with a graph that curves upward. As x increases, the slope of the tangent line increases. Thus, since the derivative increases as x increases, f ′ is an increasing function. We say this function f is concave up. Figure 4.34(b) shows a function f that curves downward. As x increases, the slope of the tangent line decreases. Since the derivative decreases as x increases, f ′ is a decreasing function. We say this function f is concave down. Definition Let f be a function that is differentiable over an open interval I. If f ′ is increasing over I, we say f is concave up over I. If f ′ is decreasing over I, we say f is concave down over I. Figure 4.34 (a), (c) Since f ′ is increasing over the interval (a, b), we say f is concave up over (a, b). (b), (d) Since f ′ is decreasing over the interval (a, b), we say f is concave down over (a, b). 402 Chapter 4 | Applications of Derivatives In general, without having the graph of a function f , how can we determine its concavity? By definition, a function f is concave up if f ′ is increasing.
    [Show full text]
  • 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
    [Show full text]
  • Matrix Calculus
    Appendix D Matrix Calculus From too much study, and from extreme passion, cometh madnesse. Isaac Newton [205, §5] − D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a differentiable real function f(x) : RK R with respect to its vector argument is defined uniquely in terms of partial derivatives→ ∂f(x) ∂x1 ∂f(x) , ∂x2 RK f(x) . (2053) ∇ . ∈ . ∂f(x) ∂xK while the second-order gradient of the twice differentiable real function with respect to its vector argument is traditionally called the Hessian; 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂x1 ∂x1∂x2 ··· ∂x1∂xK 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 2 K f(x) , ∂x2∂x1 ∂x2 ··· ∂x2∂xK S (2054) ∇ . ∈ . .. 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂xK ∂x1 ∂xK ∂x2 ∂x ··· K interpreted ∂f(x) ∂f(x) 2 ∂ ∂ 2 ∂ f(x) ∂x1 ∂x2 ∂ f(x) = = = (2055) ∂x1∂x2 ³∂x2 ´ ³∂x1 ´ ∂x2∂x1 Dattorro, Convex Optimization Euclidean Distance Geometry, Mεβoo, 2005, v2020.02.29. 599 600 APPENDIX D. MATRIX CALCULUS The gradient of vector-valued function v(x) : R RN on real domain is a row vector → v(x) , ∂v1(x) ∂v2(x) ∂vN (x) RN (2056) ∇ ∂x ∂x ··· ∂x ∈ h i while the second-order gradient is 2 2 2 2 , ∂ v1(x) ∂ v2(x) ∂ vN (x) RN v(x) 2 2 2 (2057) ∇ ∂x ∂x ··· ∂x ∈ h i Gradient of vector-valued function h(x) : RK RN on vector domain is → ∂h1(x) ∂h2(x) ∂hN (x) ∂x1 ∂x1 ··· ∂x1 ∂h1(x) ∂h2(x) ∂hN (x) h(x) , ∂x2 ∂x2 ··· ∂x2 ∇ .
    [Show full text]
  • Calculus 120, Section 7.3 Maxima & Minima of Multivariable Functions
    Calculus 120, section 7.3 Maxima & Minima of Multivariable Functions notes by Tim Pilachowski A quick note to start: If you’re at all unsure about the material from 7.1 and 7.2, now is the time to go back to review it, and get some help if necessary. You’ll need all of it for the next three sections. Just like we had a first-derivative test and a second-derivative test to maxima and minima of functions of one variable, we’ll use versions of first partial and second partial derivatives to determine maxima and minima of functions of more than one variable. The first derivative test for functions of more than one variable looks very much like the first derivative test we have already used: If f(x, y, z) has a relative maximum or minimum at a point ( a, b, c) then all partial derivatives will equal 0 at that point. That is ∂f ∂f ∂f ()a b,, c = 0 ()a b,, c = 0 ()a b,, c = 0 ∂x ∂y ∂z Example A: Find the possible values of x, y, and z at which (xf y,, z) = x2 + 2y3 + 3z 2 + 4x − 6y + 9 has a relative maximum or minimum. Answers : (–2, –1, 0); (–2, 1, 0) Example B: Find the possible values of x and y at which fxy(),= x2 + 4 xyy + 2 − 12 y has a relative maximum or minimum. Answer : (4, –2); z = 12 Notice that the example above asked for possible values. The first derivative test by itself is inconclusive. The second derivative test for functions of more than one variable is a good bit more complicated than the one used for functions of one variable.
    [Show full text]
  • Summary of Derivative Tests
    Summary of Derivative Tests Note that for all the tests given below it is assumed that the function f is continuous. Critical Numbers Definition. A critical number of a function f is a number c in the domain of f such that either f 0(c) = 0 or f 0(c) does not exist. Critical numbers tell you where a possible max/min occurs. Closed Interval Method Use: To find absolute mins/maxes on an interval [a; b] Steps: 1. Find the critical numbers of f [set f 0(x) = 0 and solve] 2. Find the value of f at each critical number 3. Find the value of f at each endpoint [i.e., find f(a) and f(b)] 4. Compare all of the above function values. The largest one is the absolute max and the smallest one is the absolute min. First Derivative Test Use: To find local max/mins Statement of the test: Suppose c is a critical number. 1. If f 0 changes from positive to negative at c; then f has a local max at c 2. If f 0 changes from negative to positive at c, then f has a local min at c 3. If f 0 is positive to the left and right of c [or negative to the left and right of c], then f has no local max or min at c Trick to Remember: If you are working on a problem and forget what means what in the above test, draw a picture of tangent lines around a minimum or around a maximum 1 Steps: 1.
    [Show full text]
  • 440 Geophysics: Brief Notes on Math Thorsten Becker, University of Southern California, 02/2005
    440 Geophysics: Brief notes on math Thorsten Becker, University of Southern California, 02/2005 Linear algebra Linear (vector) algebra is discussed in the appendix of Fowler and in your favorite math textbook. I only make brief comments here, to clarify and summarize a few issues that came up in class. At places, these are not necessarily mathematically rigorous. Dot product We have made use of the dot product, which is defined as n c =~a ·~b = ∑ aibi, (1) i=1 where ~a and~b are vectors of dimension n (n-dimensional, pointed objects like a velocity) and the n outcome of this operation is a scalar (a regular number), c. In eq. (1), ∑i=1 means “sum all that follows while increasing the index i from the lower limit, i = 1, in steps of of unity, to the upper limit, i = n”. In the examples below, we will assume a typical, spatial coordinate system with n = 3 so that ~a ·~b = a1b1 + a2b2 + a3b3, (2) where 1, 2, 3 refer to the vectors components along x, y, and z axis, respectively. When we write out the vector components, we put them on top of each other a1 ax ~a = a2 = ay (3) a3 az or in a list, maybe with curly brackets, like so: ~a = {a1,a2,a3}. On the board, I usually write vectors as a rather than ~a, because that’s easier. You may also see vectors printed as bold face letters, like so: a. We can write the amplitude or length of a vector as s n q q 2 2 2 2 2 2 2 |~a| = ∑ai = a1 + a2 + a3 = ax + ay + az .
    [Show full text]
  • Just the Definitions, Theorem Statements, Proofs On
    MATH 3333{INTERMEDIATE ANALYSIS{BLECHER NOTES 55 Abbreviated notes version for Test 3: just the definitions, theorem statements, proofs on the list. I may possibly have made a mistake, so check it. Also, this is not intended as a REPLACEMENT for your classnotes; the classnotes have lots of other things that you may need for your understanding, like worked examples. 6. The derivative 6.1. Differentiation rules. Definition: Let f :(a; b) ! R and c 2 (a; b). If the limit lim f(x)−f(c) exists and is finite, then we say that f is differentiable at x!c x−c 0 df c, and we write this limit as f (c) or dx (c). This is the derivative of f at c, and also obviously equals lim f(c+h)−f(c) by setting x = c + h or h = x − c. If f is h!0 h differentiable at every point in (a; b), then we say that f is differentiable on (a; b). Theorem 6.1. If f :(a; b) ! R is differentiable at at a point c 2 (a; b), then f is continuous at c. Proof. If f is differentiable at c then f(x) − f(c) f(x) = (x − c) + f(c) ! f 0(c)0 + f(c) = f(c); x − c as x ! c. So f is continuous at c. Theorem 6.2. (Calculus I differentiation laws) If f; g :(a; b) ! R is differentiable at a point c 2 (a; b), then (1) f(x) + g(x) is differentiable at c and (f + g)0(c) = f 0(c) + g0(c).
    [Show full text]