Function Composition and Chain Rules

Size: px
Start display at page:

Download "Function Composition and Chain Rules"

Transcription

1 Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017

2 Outline 1 Function Composition and Continuity 2

3 Function Composition and Continuity Te composition of functions is actually a simple concept. You sove one function into anoter and calculate te result.

4 Function Composition and Continuity Te composition of functions is actually a simple concept. You sove one function into anoter and calculate te result. We know ow to find x 2 and u 2 for any x and u. So wat about (x 2 + 3) 2? Tis just means take u = x and square it.

5 Function Composition and Continuity Te composition of functions is actually a simple concept. You sove one function into anoter and calculate te result. We know ow to find x 2 and u 2 for any x and u. So wat about (x 2 + 3) 2? Tis just means take u = x and square it. Tat is if f (u) = u 2 and g(x) = x 2 + 3, te composition of f and g is simply f (g(x)). Let s talk about continuity first and ten we ll go on to te idea of taking te derivative of a composition. If f and g are bot continuous, ten if you tink about it, anoter way to prase te continuity of f is tat lim y x f (y) = f (x) = f (x) = f ( lim y x y).

6 Function Composition and Continuity Te composition of functions is actually a simple concept. You sove one function into anoter and calculate te result. We know ow to find x 2 and u 2 for any x and u. So wat about (x 2 + 3) 2? Tis just means take u = x and square it. Tat is if f (u) = u 2 and g(x) = x 2 + 3, te composition of f and g is simply f (g(x)). Let s talk about continuity first and ten we ll go on to te idea of taking te derivative of a composition. If f and g are bot continuous, ten if you tink about it, anoter way to prase te continuity of f is tat lim y x f (y) = f (x) = f (x) = f ( lim y x y). So if f and g are bot continuous, we can say lim y x f (g(y)) = f ( lim g(y)) = f (g( lim y)) = f (g(x)). y x y x

7 Function Composition and Continuity Te composition of functions is actually a simple concept. You sove one function into anoter and calculate te result. We know ow to find x 2 and u 2 for any x and u. So wat about (x 2 + 3) 2? Tis just means take u = x and square it. Tat is if f (u) = u 2 and g(x) = x 2 + 3, te composition of f and g is simply f (g(x)). Let s talk about continuity first and ten we ll go on to te idea of taking te derivative of a composition. If f and g are bot continuous, ten if you tink about it, anoter way to prase te continuity of f is tat lim y x f (y) = f (x) = f (x) = f ( lim y x y). So if f and g are bot continuous, we can say lim y x f (g(y)) = f ( lim g(y)) = f (g( lim y)) = f (g(x)). y x y x So te composition of continuous functions is continuous. Heave a big sig of relief as smootness as not been lost by pusing one smoot function into anoter smoot function!

8 Function Composition and Continuity Let s do tis using te ɛ δ approac. We assume f is locally defined at te point g(x) and g is locally defined at te point x. So tere are radii r f and r g so tat f (w) is defined if w B rf (g(x)) and g(y) is defined if y B rg (x). For any ɛ, tere is a δ 1 so tat f (u) f (g(x)) < ɛ if u g(x) < δ 1 because f is continuous at x. Of course, δ 1 < r f. For te tolerance δ 1, tere is a δ 2 so tat g(y) g(x)) < δ 1 if y x < δ 2 because f is continuous at x. Of course, δ 2 < r g. Tus, y x < δ 2 implies g(y) g(x)) < δ 1 wic implies f (g(y)) f (g(x)) < ɛ. Tis sows f g is continuous at x. You sould be able to understand bot types of arguments ere!

9 Te next question is weter or not te composition of functions aving derivatives gives a new function wic as a derivative. And ow could we calculate tis derivative if te answer is yes? It turns out tis is true but to see if requires a bit more work wit limits.

10 Te next question is weter or not te composition of functions aving derivatives gives a new function wic as a derivative. And ow could we calculate tis derivative if te answer is yes? It turns out tis is true but to see if requires a bit more work wit limits. So we want to know wat d dx (f (g(x))) is. First, if g was always constant, te answer is easy. It is d dx (f (constant)) = 0 wic is a special case of te formula we are going to develop.

11 Te next question is weter or not te composition of functions aving derivatives gives a new function wic as a derivative. And ow could we calculate tis derivative if te answer is yes? It turns out tis is true but to see if requires a bit more work wit limits. So we want to know wat d dx (f (g(x))) is. First, if g was always constant, te answer is easy. It is d dx (f (constant)) = 0 wic is a special case of te formula we are going to develop. So let s assume g is not constant locally. So tere is a radius r > 0 so tat g(y) g(x) for all y B r (x). Anoter way of saying tis is g(x + ) g(x) 0 if < r. Ten, we want to calculate d (f (g(x))) = lim dx 0 f (g(x + )) f (g(x)). Rewrite by dividing and multiplying by g(x + ) g(x) wic is ok to do as we assume g is not constant locally and so we don t divide by 0. We get ( f (g(x))) = lim 0 f (g(x + )) f (g(x)) g(x + ) g(x) g(x + ) g(x)

12 g(x+) g(x) Now lim 0 = g (x) because we know g is differentiable at x.

13 g(x+) g(x) Now lim 0 = g (x) because we know g is differentiable at x. Now let u = g(x) and = g(x + ) g(x). Ten, g(x + ) = g(x) + = u +. Ten, we ave f (g(x+)) f (g(x)) g(x+) g(x) = f (u+) f (u). Since g is continuous because g as a derivative, as lim 0 = lim 0 (g(x + ) g(x)) = 0. Also, since f( is differentiable at ) u = g(x), we know lim 0 f (u + ) f (u) / = f (u) = f (g(x)).

14 g(x+) g(x) Now lim 0 = g (x) because we know g is differentiable at x. Now let u = g(x) and = g(x + ) g(x). Ten, g(x + ) = g(x) + = u +. Ten, we ave f (g(x+)) f (g(x)) g(x+) g(x) = f (u+) f (u). Since g is continuous because g as a derivative, as lim 0 = lim 0 (g(x + ) g(x)) = 0. Also, since f( is differentiable at ) u = g(x), we know lim 0 f (u + ) f (u) / = f (u) = f (g(x)). Tus, lim 0 = lim 0 f (g(x + )) f (g(x)) g(x + ) g(x) f (u + ) f (u) = lim 0 f (u + ) f (u) = f (u) = f (g(x)).

15 Since bot limits above exist, we know ( f (g(x + )) f (g(x)) lim 0 g(x + ) g(x) ( lim 0 f (g(x + )) f (g(x)) g(x + ) g(x) Tis result is called te. ) g(x + ) g(x) = ) ( g(x + ) g(x) lim 0 ) = f (g(x) g (x). Teorem For Differentiation If te composition of f and g is locally defined at a number x and if bot f (g(x)) and g (x) exist, ten te derivative of te composition of f and g also exists and is given by d dx (f (g(x)) = f (g(x)) g (x) Proof We reasoned tis out above. We usually tink about tis as follows d dx (f (inside)) = f (inside) inside (x).

16 Let s do a error term based proof. Teorem For Differentiation If te composition of f and g is locally defined at a number x and if bot f (g(x)) and g (x) exist, ten te derivative of te composition of f and g also exists and is given by d dx (f (g(x)) = f (g(x)) g (x) Proof Again, we do te case were g(x) is not a constant locally. Like before, let = g(x + ) g(x) wit u = g(x). Using error terms, since f as a derivative at u = g(x) and g as a derivative at x, we can write f (u + ) = f (u) + f (u) + E f (u +, u) were lim 0 of bot E f (u +, u) and E f (u +, u)/ are zero and g(x + ) = g(x) + g (x) + E g (x +, x) were lim 0 of bot E g (x +, x) and E g (x +, x)/ are zero.

17 Proof Ten, since u = g(x) and = g(x + ) g(x), we ave d (f (g(x))) = lim dx 0 f (g(x + )) f (g(x)) f (u + ) f (u) = lim 0 f (u) + E f (u +, u) = lim 0 ( f (u) + E f (u +, u) = lim 0 ) were we know 0 locally because g is not constant locally.

18 Proof We see lim 0 derivative at x. = lim 0 g(x+) g(x) Next, lim 0 f (u) +E f (u+,u) = g (x) because g as a = f E (u) + lim f (u+,u) 0. But E f (u +, u) lim 0 E f (u +, u) = lim = 0. 0 f Tus, lim (u) +E f (u+,u) 0 ( d (f (g(x))) = dx = f (u). Since bot limits exist, we ave lim (f (u) + E f (u +, u) 0 = f (g(x)) g (x). ) ( lim 0 )

19 Let s do an ɛ δ based proof. Teorem For Differentiation If te composition of f and g is locally defined at a number x and if bot f (g(x)) and g (x) exist, ten te derivative of te composition of f and g also exists and is given by d dx (f (g(x)) = f (g(x)) g (x) Proof Again, we do te case were g(x) is not a constant locally. Like before, let = g(x + ) g(x) wit u = g(x). Since g is not constant locally at x, tere is a radius r g so tat g(x ) g(x) 0 wen < r g. Again we write d (f (g(x))) = lim dx 0 f (g(x + )) f (g(x)) g(x + ) g(x) g(x + ) g(x)

20 Proof Ten, since u = g(x) and = g(x + ) g(x), we ave f (g(x + )) f (g(x)) g(x + ) g(x) g(x + ) g(x) = f (u + ) f (u). Since g is differentiable at x, for a given ξ 1, tere is a δ 1 so tat < δ 1 implies g (x) ξ 1 < g(x+) g(x) = < g (x) + ξ 1. And since f is differentiable at u = g(x), for a given ξ 2, tere is a δ 2 so tat < δ 2 implies f f (u+) f (u) (u) ξ 2 < < f (u) + ξ 2. f (g(x+)) f (g(x)) g(x+) g(x) g(x+) g(x) Let for convenience. Ten, if δ < min{r g, δ 1, δ 2 }, all conditions old and we ave = f (f (u) ξ 2 ) (g (x) ξ 1 ) < f < (f (u) + ξ 2 ) (g (x) + ξ 1 )

21 Proof Multiplying out tese terms and canceling te ξ 1 ξ 2, we ave or f (u)g (x) ξ 1 f (u) ξ 2 g (x) < f < f (u)g (x) + ξ 1 f (u) + ξ 2 g (x) ξ 1 f (u) ξ 2 g (x) < f Tus, ( f (u)g (x) ) < ξ 1 f (u) + ξ 2 g (x) f ( f (u)g (x) ) < ξ 1 f (u) + ξ 2 g (x)

22 Proof Coose ξ 1 = ɛ 2( f (g(x)) +1) and ξ 2 = Ten, we ave < δ implies f Tis proves te result! ɛ 2( g (x) +1). ( f (u)g (x) ) < ɛ. Comment You sould know ow to attack tis proof all tree ways!

23 Example Find te derivative of (t 3 + 4) 3. Solution It is easy to do tis if we tink about it tis way. ( (ting) power ) = power (ting) power - 1 (ting) Tus, ( (t 3 + 4) 3) = 3 (t 3 + 4) 2 (3t 2 )

24 Example Find te derivative of 1/(t 2 + 4) 3. Solution Tis is also ( (ting) power ) = power (ting) power - 1 (ting) were power is 3 and ting is t So we get ( 1/(t 2 + 4) 3) = 3 (t 2 + 4) 4 (2t)

25 Example Find te derivative of (6t 4 + 9t 2 + 8) 6. Solution ( (6t 4 + 9t 2 + 8) 6 ) = 6 (6t 4 + 9t 2 + 8) 5 (24t t)

26 Homework Use an ɛ δ argument to sow 7f (x) + 2g(x) is differentiable at any x were f and g are differentiable Use an ɛ N argument to sow 19a n 24b n converges if (a n ) and (b n ) converge Recall in te error form for differentiation, f (x + ) = f (x) + f (x) + E(x +, x) were te linear function T (x) = f (x) + f (x) is called te tangent line approximation to f at te base point x. For f (x) = 2x 3 + 3x + 2, sketc f (x) and T (x) on te same grap carefully at te points x = 1, x = 0.5 and x = 1.3. f (x+) f (x) Also draw a sample slope triangle for eac base point. Finally, draw in te error function as a vertical line at te base points. Use multiple colors!! 23.4 If f is continuous at x, wy is it true tat lim n f (x n ) = f (x) for any sequence (x n ) wit x n x?

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition an Cain Rules James K. Peterson Department of Biological Sciences an Department of Matematical Sciences Clemson University November 2, 2018 Outline Function Composition an Continuity

More information

Sin, Cos and All That

Sin, Cos and All That Sin, Cos and All Tat James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 9, 2017 Outline Sin, Cos and all tat! A New Power Rule Derivatives

More information

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as

More information

Exam 1 Review Solutions

Exam 1 Review Solutions Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),

More information

Continuity and Differentiability Worksheet

Continuity and Differentiability Worksheet Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

More information

. Compute the following limits.

. Compute the following limits. Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim

More information

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0 3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,

More information

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example, NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing

More information

Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.

Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point. Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope

More information

1 Lecture 13: The derivative as a function.

1 Lecture 13: The derivative as a function. 1 Lecture 13: Te erivative as a function. 1.1 Outline Definition of te erivative as a function. efinitions of ifferentiability. Power rule, erivative te exponential function Derivative of a sum an a multiple

More information

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a? Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is

More information

Calculus I Homework: The Derivative as a Function Page 1

Calculus I Homework: The Derivative as a Function Page 1 Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.

More information

2.11 That s So Derivative

2.11 That s So Derivative 2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point

More information

MVT and Rolle s Theorem

MVT and Rolle s Theorem AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state

More information

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225 THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:

More information

How to Find the Derivative of a Function: Calculus 1

How to Find the Derivative of a Function: Calculus 1 Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te

More information

Introduction to Derivatives

Introduction to Derivatives Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))

More information

Precalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!

Precalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here! Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.

More information

f a h f a h h lim lim

f a h f a h h lim lim Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point

More information

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t). . Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use

More information

Lesson 6: The Derivative

Lesson 6: The Derivative Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange

More information

Derivatives of Exponentials

Derivatives of Exponentials mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function

More information

Combining functions: algebraic methods

Combining functions: algebraic methods Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

More information

1 Solutions to the in class part

1 Solutions to the in class part NAME: Solutions to te in class part. Te grap of a function f is given. Calculus wit Analytic Geometry I Exam, Friday, August 30, 0 SOLUTIONS (a) State te value of f(). (b) Estimate te value of f( ). (c)

More information

Practice Problem Solutions: Exam 1

Practice Problem Solutions: Exam 1 Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical

More information

REVIEW LAB ANSWER KEY

REVIEW LAB ANSWER KEY REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g

More information

2.3 Algebraic approach to limits

2.3 Algebraic approach to limits CHAPTER 2. LIMITS 32 2.3 Algebraic approac to its Now we start to learn ow to find its algebraically. Tis starts wit te simplest possible its, and ten builds tese up to more complicated examples. Fact.

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note

More information

4.2 - Richardson Extrapolation

4.2 - Richardson Extrapolation . - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence

More information

1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 -

1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 - Eercise. Find te derivative of g( 3 + t5 dt wit respect to. Solution: Te integrand is f(t + t 5. By FTC, f( + 5. Eercise. Find te derivative of e t2 dt wit respect to. Solution: Te integrand is f(t e t2.

More information

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit

More information

3.4 Worksheet: Proof of the Chain Rule NAME

3.4 Worksheet: Proof of the Chain Rule NAME Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are

More information

HOMEWORK HELP 2 FOR MATH 151

HOMEWORK HELP 2 FOR MATH 151 HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,

More information

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ). Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate

More information

MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM

MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM MA9-A Applied Calculus for Business 006 Fall Homework Solutions Due 9/9/006 0:0AM. #0 Find te it 5 0 + +.. #8 Find te it. #6 Find te it 5 0 + + = (0) 5 0 (0) + (0) + =.!! r + +. r s r + + = () + 0 () +

More information

Some Review Problems for First Midterm Mathematics 1300, Calculus 1

Some Review Problems for First Midterm Mathematics 1300, Calculus 1 Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,

More information

University Mathematics 2

University Mathematics 2 University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at

More information

Main Points: 1. Limit of Difference Quotients. Prep 2.7: Derivatives and Rates of Change. Names of collaborators:

Main Points: 1. Limit of Difference Quotients. Prep 2.7: Derivatives and Rates of Change. Names of collaborators: Name: Section: Names of collaborators: Main Points:. Definition of derivative as limit of difference quotients. Interpretation of derivative as slope of grap. Interpretation of derivative as instantaneous

More information

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4. December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need

More information

Higher Derivatives. Differentiable Functions

Higher Derivatives. Differentiable Functions Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.

More information

7.1 Using Antiderivatives to find Area

7.1 Using Antiderivatives to find Area 7.1 Using Antiderivatives to find Area Introduction finding te area under te grap of a nonnegative, continuous function f In tis section a formula is obtained for finding te area of te region bounded between

More information

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014 Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.

More information

A.P. CALCULUS (AB) Outline Chapter 3 (Derivatives)

A.P. CALCULUS (AB) Outline Chapter 3 (Derivatives) A.P. CALCULUS (AB) Outline Capter 3 (Derivatives) NAME Date Previously in Capter 2 we determined te slope of a tangent line to a curve at a point as te limit of te slopes of secant lines using tat point

More information

Lab 6 Derivatives and Mutant Bacteria

Lab 6 Derivatives and Mutant Bacteria Lab 6 Derivatives and Mutant Bacteria Date: September 27, 20 Assignment Due Date: October 4, 20 Goal: In tis lab you will furter explore te concept of a derivative using R. You will use your knowledge

More information

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2

More information

Section 3: The Derivative Definition of the Derivative

Section 3: The Derivative Definition of the Derivative Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding te slope

More information

Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of

More information

1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.

1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits. Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te

More information

INTRODUCTION TO CALCULUS LIMITS

INTRODUCTION TO CALCULUS LIMITS Calculus can be divided into two ke areas: INTRODUCTION TO CALCULUS Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and

More information

Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 =

Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 = Test Review Find te determinant of te matrix below using (a cofactor expansion and (b row reduction Answer: (a det + = (b Observe R R R R R R R R R Ten det B = (((det Hence det Use Cramer s rule to solve:

More information

The Derivative The rate of change

The Derivative The rate of change Calculus Lia Vas Te Derivative Te rate of cange Knowing and understanding te concept of derivative will enable you to answer te following questions. Let us consider a quantity wose size is described by

More information

Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser

More information

1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist

1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist Mat 1120 Calculus Test 2. October 18, 2001 Your name Te multiple coice problems count 4 points eac. In te multiple coice section, circle te correct coice (or coices). You must sow your work on te oter

More information

5. (a) Find the slope of the tangent line to the parabola y = x + 2x

5. (a) Find the slope of the tangent line to the parabola y = x + 2x MATH 141 090 Homework Solutions Fall 00 Section.6: Pages 148 150 3. Consider te slope of te given curve at eac of te five points sown (see text for figure). List tese five slopes in decreasing order and

More information

Gradient Descent etc.

Gradient Descent etc. 1 Gradient Descent etc EE 13: Networked estimation and control Prof Kan) I DERIVATIVE Consider f : R R x fx) Te derivative is defined as d fx) = lim dx fx + ) fx) Te cain rule states tat if d d f gx) )

More information

MAT 1339-S14 Class 2

MAT 1339-S14 Class 2 MAT 1339-S14 Class 2 July 07, 2014 Contents 1 Rate of Cange 1 1.5 Introduction to Derivatives....................... 1 2 Derivatives 5 2.1 Derivative of Polynomial function.................... 5 2.2 Te

More information

Derivatives. By: OpenStaxCollege

Derivatives. By: OpenStaxCollege By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator

More information

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator. Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative

More information

Math 1241 Calculus Test 1

Math 1241 Calculus Test 1 February 4, 2004 Name Te first nine problems count 6 points eac and te final seven count as marked. Tere are 120 points available on tis test. Multiple coice section. Circle te correct coice(s). You do

More information

Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and the Gradient Vector Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

More information

158 Calculus and Structures

158 Calculus and Structures 58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you

More information

We name Functions f (x) or g(x) etc.

We name Functions f (x) or g(x) etc. Section 2 1B: Function Notation Bot of te equations y 2x +1 and y 3x 2 are functions. It is common to ave two or more functions in terms of x in te same problem. If I ask you wat is te value for y if x

More information

Differentiation in higher dimensions

Differentiation in higher dimensions Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends

More information

Math 161 (33) - Final exam

Math 161 (33) - Final exam Name: Id #: Mat 161 (33) - Final exam Fall Quarter 2015 Wednesday December 9, 2015-10:30am to 12:30am Instructions: Prob. Points Score possible 1 25 2 25 3 25 4 25 TOTAL 75 (BEST 3) Read eac problem carefully.

More information

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim

More information

Math 312 Lecture Notes Modeling

Math 312 Lecture Notes Modeling Mat 3 Lecture Notes Modeling Warren Weckesser Department of Matematics Colgate University 5 7 January 006 Classifying Matematical Models An Example We consider te following scenario. During a storm, a

More information

Integral Calculus, dealing with areas and volumes, and approximate areas under and between curves.

Integral Calculus, dealing with areas and volumes, and approximate areas under and between curves. Calculus can be divided into two ke areas: Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and minima problems Integral

More information

MATH CALCULUS I 2.1: Derivatives and Rates of Change

MATH CALCULUS I 2.1: Derivatives and Rates of Change MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main

More information

(4.2) -Richardson Extrapolation

(4.2) -Richardson Extrapolation (.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as

More information

The derivative function

The derivative function Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative

More information

Time (hours) Morphine sulfate (mg)

Time (hours) Morphine sulfate (mg) Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15

More information

3.1 Extreme Values of a Function

3.1 Extreme Values of a Function .1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find

More information

4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.

4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these. Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra

More information

Section 3.1: Derivatives of Polynomials and Exponential Functions

Section 3.1: Derivatives of Polynomials and Exponential Functions Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 6. Differential Calculus 6.. Differentiation from First Principles. In tis capter, we will introduce

More information

1 Limits and Continuity

1 Limits and Continuity 1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function

More information

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx. Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions

More information

232 Calculus and Structures

232 Calculus and Structures 3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE

More information

Finding and Using Derivative The shortcuts

Finding and Using Derivative The shortcuts Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex

More information

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x

More information

Mathematics 123.3: Solutions to Lab Assignment #5

Mathematics 123.3: Solutions to Lab Assignment #5 Matematics 3.3: Solutions to Lab Assignment #5 Find te derivative of te given function using te definition of derivative. State te domain of te function and te domain of its derivative..: f(x) 6 x Solution:

More information

2.3 Product and Quotient Rules

2.3 Product and Quotient Rules .3. PRODUCT AND QUOTIENT RULES 75.3 Product and Quotient Rules.3.1 Product rule Suppose tat f and g are two di erentiable functions. Ten ( g (x)) 0 = f 0 (x) g (x) + g 0 (x) See.3.5 on page 77 for a proof.

More information

Definition of the Derivative

Definition of the Derivative Te Limit Definition of te Derivative Tis Handout will: Define te limit grapically and algebraically Discuss, in detail, specific features of te definition of te derivative Provide a general strategy of

More information

Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if

Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions

More information

Differential Calculus (The basics) Prepared by Mr. C. Hull

Differential Calculus (The basics) Prepared by Mr. C. Hull Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit

More information

MTH-112 Quiz 1 Name: # :

MTH-112 Quiz 1 Name: # : MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation

More information

1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)

1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x) Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of

More information

Continuity and Differentiability of the Trigonometric Functions

Continuity and Differentiability of the Trigonometric Functions [Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te

More information

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) *

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) * OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) * Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

More information

2.1 THE DEFINITION OF DERIVATIVE

2.1 THE DEFINITION OF DERIVATIVE 2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative

More information

Numerical Differentiation

Numerical Differentiation Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function

More information

Mathematics 105 Calculus I. Exam 1. February 13, Solution Guide

Mathematics 105 Calculus I. Exam 1. February 13, Solution Guide Matematics 05 Calculus I Exam February, 009 Your Name: Solution Guide Tere are 6 total problems in tis exam. On eac problem, you must sow all your work, or oterwise torougly explain your conclusions. Tere

More information

1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2

1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2 MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are

More information

Exponentials and Logarithms Review Part 2: Exponentials

Exponentials and Logarithms Review Part 2: Exponentials Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power

More information

Copyright c 2008 Kevin Long

Copyright c 2008 Kevin Long Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )

More information

Rules of Differentiation

Rules of Differentiation LECTURE 2 Rules of Differentiation At te en of Capter 2, we finally arrive at te following efinition of te erivative of a function f f x + f x x := x 0 oing so only after an extene iscussion as wat te

More information

SFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00

SFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00 SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only

More information

1watt=1W=1kg m 2 /s 3

1watt=1W=1kg m 2 /s 3 Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory

More information