Basic Differentiation Rules and Rates of Change the Constant Rule the Derivative of a Constant Function Is 0

Copy Link

Basic Differentiation Rules and Rates of Change The Constant Rule The derivative of a constant function is 0. For any real number, c y The slope of a horizontal line is 0. x The derivative of a constant function is 0. 1 PROOF: Let f(x) = c, Then by the limit definition of the derivative, EX. #1 Using the Constant Rule Function Derivative 2 The Power Rule The derivative of the term axn , where a and n are real numbers, is STEPS: 1. Multiply the coefficient by the variable's exponent. If no coefficient is stated – in other words, the coefficient equals 1– the exponent becomes the new coefficient. 2. Subtract 1 from the exponent. EX.#2: Use the power rule to find f '(x) if: 3 Finding the Slope of a Graph EX.#3: Find the slope of the graph of f(x) when: The slope of a graph at a point is the value of the derivative at that point. a. x = 2 b. x = 0 c. x = 2 4 Finding an Equation of a Tangent Line EX.#4: Find the equation of the tangent line to the graph of f(x) when: The derivative of f is: a. At x = 2, y ' = 3 b. At x = 0, y ' = 1 c. At x = 2 , y ' = 5 5 Derivatives of the Sine and Cosine Functions EX #5: Derivatives Involving Sines and Cosines Function Derivative 6 RATES OF CHANGE The derivative can determine slope and can also be used to determine the rate of change of one variable with respect to another. It is customary to describe the motion of an object moving in a straight line with either a horizontal or vertical line, from some designated origin, to represent the line of motion. • Movement right or upward is considered positive direction. • Movement left or downward is considered negative direction. Vocabulary rate of change derivative instantaneous rate of change } displacement distance traveled 7 FORMULAS TO RECOGNIZE General Position Function g gravitational constant v0 initial velocity s0 initial height Free Fall Constants English Metric – 32 ft/sec2 – 9.8 m/sec2 Velocity Function 1. Tells how fast an object is moving 2. Tells the direction of motion object moves forward ⇒positive object moves backward ⇒ negative 3. Velocity is derivative of position function with respect to time. Average Velocity of an Object over time Interval Distance Rate = Time Change in distance V = avg Change in time 8 The average velocity between P t1 and t2 is the slope of the secant line, and the instantaneous velocity at t1 is the slope of the tangent line. t1 = 1 t2 9 Speed 1. absolute value of velocity 2. measures the rate of motion, regardless of direction 3. nonnegative velocity Speed = Acceleration 1. rate at which velocity changes 2. measures how quickly the body picks up or loses speed. 3. derivative of velocity or second derivative of position function. 10 EX #6: A coin is dropped from the top of a building that is 1362 feet tall. A. Write the position function. B. Write the velocity function. C. Find the average velocity on [1, 2]. Vavg = 11 D. Find instantaneous velocities when t = 1 t = 2 E. When will coin reach ground? F. What is the velocity at impact? G. Convert (f) to miles per hour. 12.

Recommended publications

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.
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.
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.
Section 9.6, the Chain Rule and the Power Rule

Section 9.6, The Chain Rule and the Power Rule Chain Rule: If f and g are differentiable functions with y = f(u) and u = g(x) (i.e. y = f(g(x))), then dy = f 0(u) · g0(x) = f 0(g(x)) · g0(x); or dx dy dy du = · dx du dx For now, we will only be considering a special case of the Chain Rule. When f(u) = un, this is called the (General) Power Rule. (General) Power Rule: If y = un, where u is a function of x, then dy du = nun−1 · dx dx Examples Calculate the derivatives for the following functions: 1. f(x) = (2x + 1)7 Here, u(x) = 2x + 1 and n = 7, so f 0(x) = 7u(x)6 · u0(x) = 7(2x + 1)6 · (2) = 14(2x + 1)6. 2. g(z) = (4z2 − 3z + 4)9 We will take u(z) = 4z2 − 3z + 4 and n = 9, so g0(z) = 9(4z2 − 3z + 4)8 · (8z − 3). 1 2 −2 3. y = (x2−4)2 = (x − 4) Letting u(x) = x2 − 4 and n = −2, y0 = −2(x2 − 4)−3 · 2x = −4x(x2 − 4)−3. p 4. f(x) = 3 1 − x2 + x4 = (1 − x2 + x4)1=3 0 1 2 4 −2=3 3 f (x) = 3 (1 − x + x ) (−2x + 4x ). Notes for the Chain and (General) Power Rules: 1. If you use the u-notation, as in the definition of the Chain and Power Rules, be sure to have your final answer use the variable in the given problem.
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.
4.1 AP Calc Antiderivatives and Indefinite Integration.Notebook November 28, 2016

4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 4.1 Antiderivatives and Indefinite Integration Learning Targets 1. Write the general solution of a differential equation 2. Use indefinite integral notation for antiderivatives 3. Use basic integration rules to find antiderivatives 4. Find a particular solution of a differential equation 5. Plot a slope field 6. Working backwards in physics Intro/Warmup 1 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Note: I am changing up the procedure of the class! When you walk in the class, you will put a tally mark on those problems that you would like me to go over, and you will turn in your homework with your name, the section and the page of the assignment. Order of the day 2 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Vocabulary First, find a function F whose derivative is . because any other function F work? So represents the family of all antiderivatives of . The constant C is called the constant of integration. The family of functions represented by F is the general antiderivative of f, and is the general solution to the differential equation The operation of finding all solutions to the differential equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign: is read as the antiderivative of f with respect to x. Vocabulary 3 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Nov 289:38 AM 4 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 ∫ Use basic integration rules to find antiderivatives The Power Rule: where 1.
Review of Differentiation and Integration

Schreyer Fall 2018 Review of Differentiation and Integration for Ordinary Differential Equations In this course you will be expected to be able to differentiate and integrate quickly and accurately. Many students take this course after having taken their previous course many years ago, at another institution where certain topics may have been omitted, or just feel uncomfortable with particular techniques. Because understanding this material is so important to being successful in this course, we have put together this brief review packet. In this packet you will find sample questions and a brief discussion of each topic. If you find the material in this pamphlet is not sufficient for you, it may be necessary for you to use additional resources, such as a calculus textbook or online materials. Because this is considered prerequisite material, it is ultimately your responsibility to learn it. The topics to be covered include Differentiation and Integration. 1 Differentiation Exercises: 1. Find the derivative of y = x3 sin(x). ln(x) 2. Find the derivative of y = cos(x) . 3. Find the derivative of y = ln(sin(e2x)). Discussion: It is expected that you know, without looking at a table, the following differentiation rules: d [(kx)n] = kn(kx)n−1 (1) dx d h i ekx = kekx (2) dx d 1 [ln(kx)] = (3) dx x d [sin(kx)] = k cos kx (4) dx d [cos(kx)] = −k sin x (5) dx d [uv] = u0v + uv0 (6) dx 1 d u u0v − uv0 = (7) dx v v2 d [u(v(x))] = u0(v)v0(x): (8) dx We put in the constant k into (1) - (5) because a very common mistake to make is something d e2x like: e2x = (when the correct answer is 2e2x).
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).
The Legacy of Leonhard Euler: a Tricentennial Tribute (419 Pages)

P698.TP.indd 1 9/8/09 5:23:37 PM This page intentionally left blank Lokenath Debnath The University of Texas-Pan American, USA Imperial College Press ICP P698.TP.indd 2 9/8/09 5:23:39 PM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE LEGACY OF LEONHARD EULER A Tricentennial Tribute Copyright © 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-525-0 ISBN-10 1-84816-525-0 Printed in Singapore. LaiFun - The Legacy of Leonhard.pmd 1 9/4/2009, 3:04 PM September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard Leonhard Euler (1707–1783) ii September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard To my wife Sadhana, grandson Kirin,and granddaughter Princess Maya, with love and affection.