22 mai 2017 be seen from the second derivative (if it exists). 6. Page 7. DMM summer 2017 ... Calculate the gradient of fA

optimization

15 nov. 2012 determinant derivative of inverse matrix

matrixcookbook

b − Ax2. 2. = (b − Ax)T (b − Ax). = bT b − (Ax)T b − bT Ax + xT AT Ax. = bT b − 2bT Ax + xT AT Ax Because mixed second partial derivatives satisfy.

lecture

determinant derivative of inverse matrix

Matrix Cookbook

apparent that the function you wish to integrate is a derivative in some EXAMPLE 8.1.2 Evaluate ∫ sin(ax + b) dx assuming that a and b are constants ...

calculus Techniques of Integration

Question 2: Find limx→2f(x): f(x) = 1776. (A) +∞. (B) 1770. (C) −∞. (D) Does not exist! (E) None of the above. Answer: (E) The limit of any constant

week answers

If a1b2 = a2b1 show that this equation reduces to the form y′ = g(ax + by). Solution Substituting a2 = λa1 and b2 = λb1 into the equation yields:.

Assignment Solutions

To see that action I will write b1

linearalgebra

When a differential equation contains one or more partial derivatives of an (viii) z = ax e' +. 1. 2. Sol. (1) We are given z = (2x + a) (2 y + b).

partial differential equations unit

146123

8

### Techniques of Integration

Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. For example, faced with? x 10dx we realize immediately that the derivative ofx11will supply anx10: (x11)?= 11x10. We don"t want the "11", but constants are easy to alter, becausedifferentiation "ignores" them in certain circumstances, so d dx111x11=11111x10=x10. From our knowledge of derivatives, we can immediately writedown a number of an- tiderivatives. Here is a list of those most often used: x ndx=xn+1 n+ 1+C,ifn?=-1 x -1dx= ln|x|+C e xdx=ex+C sinxdx=-cosx+C 163

#### 164Chapter 8 Techniques of Integration

cosxdx= sinx+C sec

#### 2xdx= tanx+C

secxtanxdx= secx+C ?1

#### 1 +x2dx= arctanx+C

?1 ⎷1-x2dx= arcsinx+C Needless to say, most problems we encounter will not be so simple. Here"s a slightly more complicated example: find?

#### 2xcos(x2)dx.

This is not a "simple" derivative, but a little thought reveals that it must have come from an application of the chain rule. Multiplied on the "outside" is 2x, which is the derivative of the "inside" functionx2. Checking: d dxsin(x2) = cos(x2)ddxx2= 2xcos(x2), so

#### 2xcos(x2)dx= sin(x2) +C.

Even when the chain rule has "produced" a certain derivative, it is not always easy to see. Consider this problem:? x 3?

#### 1-x2dx.

### There are two factors in this expression,x3and?

#### 1-x2, but it is not apparent that the

chain rule is involved. Some clever rearrangement reveals that it is: x 3?

#### 1-x2dx=?

(-2x)? -12? (1-(1-x2))?1-x2dx. This looks messy, but we do now have something that looks likethe result of the chain rule: the function 1-x2has been substituted into-(1/2)(1-x)⎷ x, and the derivative

#### 8.1 Substitution165

of 1-x2,-2x, multiplied on the outside. If we can find a functionF(x) whose derivative is-(1/2)(1-x)⎷ xwe"ll be done, since then d dxF(1-x2) =-2xF?(1-x2) = (-2x)? -12? (1-(1-x2))?1-x2 =x3? 1-x2

### But this isn"t hard:

-1
8

### Techniques of Integration

Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. For example, faced with? x 10dx we realize immediately that the derivative ofx11will supply anx10: (x11)?= 11x10. We don"t want the "11", but constants are easy to alter, becausedifferentiation "ignores" them in certain circumstances, so d dx111x11=11111x10=x10. From our knowledge of derivatives, we can immediately writedown a number of an- tiderivatives. Here is a list of those most often used: x ndx=xn+1 n+ 1+C,ifn?=-1 x -1dx= ln|x|+C e xdx=ex+C sinxdx=-cosx+C 163

#### 164Chapter 8 Techniques of Integration

cosxdx= sinx+C sec

#### 2xdx= tanx+C

secxtanxdx= secx+C ?1

#### 1 +x2dx= arctanx+C

?1 ⎷1-x2dx= arcsinx+C Needless to say, most problems we encounter will not be so simple. Here"s a slightly more complicated example: find?

#### 2xcos(x2)dx.

This is not a "simple" derivative, but a little thought reveals that it must have come from an application of the chain rule. Multiplied on the "outside" is 2x, which is the derivative of the "inside" functionx2. Checking: d dxsin(x2) = cos(x2)ddxx2= 2xcos(x2), so

#### 2xcos(x2)dx= sin(x2) +C.

Even when the chain rule has "produced" a certain derivative, it is not always easy to see. Consider this problem:? x 3?

#### 1-x2dx.

### There are two factors in this expression,x3and?

#### 1-x2, but it is not apparent that the

chain rule is involved. Some clever rearrangement reveals that it is: x 3?

#### 1-x2dx=?

(-2x)? -12? (1-(1-x2))?1-x2dx. This looks messy, but we do now have something that looks likethe result of the chain rule: the function 1-x2has been substituted into-(1/2)(1-x)⎷ x, and the derivative

#### 8.1 Substitution165

of 1-x2,-2x, multiplied on the outside. If we can find a functionF(x) whose derivative is-(1/2)(1-x)⎷ xwe"ll be done, since then d dxF(1-x2) =-2xF?(1-x2) = (-2x)? -12? (1-(1-x2))?1-x2 =x3? 1-x2

### But this isn"t hard:

-1