1 Chapter

0 Content 2 Background 2.1 Cosmological 2.1.11 Mathematics 2.1.11.12 Analysis

Chapter 3

Contents Chapter 1 2 3 4 5 6 7 8 9 10 - Answers 1 2 3 4 5 6 7 8 9 10 Appendix Index All

Chapter 2 Differentiation Of Functions

2.1 Calculating Derivatives Directly 
      2.1.1 Increment of the argument and increment of a function
      2.1.2 The derivative slope of the tangent 
                rate of change of the function 
      2.1.3 One-sided derivatives
      2.1.4 Infinite derivative
                Items 341 - 367  Exercises, Answers
2.2
Tables of Derivatives:
      2.2.1  Basic rules for derivatives
      2.2.2  Table of derivatives of basic functions 
      2.2.3 Rule for differentiating a compound
                function (chain rule)
                               Exercises
      A Algebraic Functions,
      B Inverse Circular and Trigonometric Functions, 
      C Exponential and Logarithmic Functions, 
      D Hyperbolic and Inverse Hyperbolic Functions, 
      E Compound Functions, 
      F Miscellaneous Functions, logarithmic derivative 
      G Logarithmic Derivative,  
  Answers
368 - 381
382 - 389
390 - 400
401 - 408
409 - 454
455 - 563
564 - 580
2.3 Derivatives of Functions Not Represented Explicitly
      2.3.1 Derivative of an inverse function
      2.3.2 The derivatives of parametrically represented functions
      2.3.3 Derivative of an implicit function
                Items 581 - 620  Exercises, Answers
2.4
Geometry and Mechanics Applications of the Derivative
      2.4.1 Equations of the tangent and the normal
      2.4.2 Angle between curves
      2.4.3 Segments associated with the tangent and the normal
                in an orthogonal co-ordinate system
     2.4.4 Segments associated with the tangent and
               the normal in a polar co-ordinate system
                Items 621 - 666  Exercises, Answers
2.5 Derivatives of Higher Orders
      2.5.1 Definition of higher derivatives
      2.5.2 Leibnitz's rule
      2.5.3 Higher-order derivatives of 
                parametrically represented functions, 
                A. Higher-order derivatives of
                     explicit functions, 
                     Items 667 - 691  Exercises, Answers
                B. Higher- Order Derivatives
                     of Parametrically Represented Functions
                     and of implicit Functions, 
                     Items 692 - 711  Exercises, Answers
2.6 Differentials of First and Higher Orders 
      2.6.1 First-order differential 
      2.6.2 Principal properties of differentials 
      2.6.3 Applying differentials to
                approximate calculations 
      2.6.4 Higher order differentials,
                Items 712 - 755  Exercises, Answers
2.7 Mean value theorems
      2.7.1 Rolle's theorem
      2.7.2 Lagrange's theorem
      2.7.3 Cauchy's theorem,
                Items 756 - 765  Exercises, Answers
2.8 Taylor's Formula, 
       Items 766 - 775  Exercises, Answers
        Maclaurin's formula   
2.9
L'Hospital - Bernoulli Rule for 
      Evaluating Indeterminate Expressions 
      2.9.1 Evaluation of the 
                indeterminate forms 0/0 and ¥/¥
   
2.9.2 Other Indeterminate forms, 
                Items 776 - 808  Exercises, Answers

2.1 Calculating Derivatives Directly  

2.1.1 Increment of the argument and increment of a function: If x and x1 are the values of the argument x and y=f(x) and y1=f(x1) are the corresponding values of the function y = f(x), then

is called the increment of the argument x in the interval (x, x1) and

is called the increment of the function y in the same interval (x, x1) (Fig, 11, where Dx = MA and Dy =AN).

The ratio

is the slope of the secant MN of the graph of the function y = f(x) and is called the mean rate of change of the function y over the interval (x, x + Dx).

Example 1. Calculate for the function

Dx and Dy. corresponding to a change in the argument: a) from x = l to x = 1.1, b) from x = 3 to x = 2.

Solution: We have

Example 2. Find for the hyperbola y = 1/x the slope of the secant passing through the points M(3, 1/3) and N ( 10, 1/10).

Solution: Here,

2.1.2 The derivative: The derivative y' of a function y = f(x) with respect to the argument x is the limit of the ratio Dy/Dx when Dx approaches zero, i.e.,

The magnitude of the derivative yields the slope of the tangent MT of the graph of the function y=f(x) at the point x (Fig,11): 

Finding the derivative y' is usually called differentiation of the function. The derivative y' = f '(x) is the rate of change of the function at the point x.

Example 3. Find the derivative of the function

Solution: By (1),

and

Hence,

2.1.3 One-sided derivatives: The expressions

are called the left-hand side and right-hand side derivative of the function f(x) at the point x, In order that f '(x) shall exist, it is necessary and sufficient that

Example 4: Find f '-(0) and f '+(0) of the function

Solution: By definition, we have

2.1.4 Infinite derivative: If at some point

we say that the continuous function f(x) has an infinite derivative at x. In this case, the tangent to the graph of the function y = f(x) is perpendicular to the x-axis.

Example 5: Find f '(0) of the function

Solution: We find

Exercises 341 - 367

341: Find the increment of the function y = x² which corresponds to a given change in argument:

342: Find Dy of the function if

343: Why can we determine for the function y=2x+3 the increment Dy, if all we know is the corresponding increment Dx=5, while for the function y = x² this cannot be done?

344: Find the increment Dy and the ratio Dy/Dx for the functions:

345. Find Dy and Dy/Dx which correspond to a change in argument from x to Dx for the functions:

346: Find the slope of the secant to the parabola

between the points:

To what limit does the slope of the secant tend in the latter case if h ® 0.

347. What Is the mean rate of change of the function y = x³ in the interval 1 £ x £ 4?

348. The law of motion of a point is s = 2t² + 3t + 5, where the distance s is in centimetres and the time t in seconds. What is the average velocity of the point over the interval of time t = 1 to t = 5?

349. Find the mean rise of the curve y = 2x in the interval 1 £ x £ 5?.

350. Find the mean rise of the curve y = f(x) in the interval [x, x + Dx].

351. What is to be understood by the rise of the curve y = f(x) at a given point x?

352. Define:

a) the mean rate of rotation;
b) the instantaneous rate of rotation.

353. A hot body, placed in a medium of lower temperature, cools off. What is to be understood by:

a) the mean rate of cooling;
b) the rate of cooling at a given instant?

354. What is to be understood by the rate of reaction of a substance in a chemical reaction?

355. Let m = f(x) be the mass of a non-homogeneous rod over the interval [0, x]. What is to be understood by:

a) the mean linear density of the rod on the interval [x, x+ Dx];
b) the linear density of the rod at a point x?

356. Find the ratio Dy/Dx of the function y = 1/x at the point x = 2, if:

a) Dx = 1, b) Dx = 0.l; c)Dx = 0.01.

What is the derivative y ' when x = 2?

357**. Find the derivative of the function y = tan x.

358. Find of the [function's:

359. Calculate f '(8) of

360. Find

361. At what points does the derivative of the. function f(x) = x³ coincide numerically with the value of the function itself, i.e., f(x) = f(x)?

362. The law of motion of a point is s = 5t², where the distance s is in metres and the time t in seconds. Find the speed at t=3.

363. Find the slope of the tangent to the curve y = 0.lx³, drawn at a point with abscissa x = 2.

364. Find the slope of the tangent to the curve y = sin x at the point (p, 0).

365. Find the value of the derivative of the function f(x) = 1/x at the point x = x0 ¹ 0.

366*. What are the slopes of the tangents to the curves y = 1/x and y = x² at the point of their intersection? Find the angle between these tangents.

367**. Show that the following functions do not have finite derivatives at the indicated points:

Hints and Answers 341 - 367

2.2 Tables of Derivatives:

2.2.1 Basic rules for derivatives: If c is a constant and u = j(x), v = y(x) are functions with derivatives, then

2.2.2 Table of derivatives of basic functions:  

2.2.3 Rule for differentiating a compound function (chain rule): If y = f(u) and = j(x), i.e., y=f[j(x)], where the .functions y and u have derivatives, then

This rule extends to a series of any finite number of differentiable functions.

Example 1. Find the derivative of the function

Solution: Setting y = u5, where u= (x² - 2x + 3), Formula (1) yields

Example 2: Find the derivative of the function

Solution: Setting

we find

Find the derivatives of the following functions (the rule for differentiating a compound function is not used in problems 368-408).

A Algebraic Functions

Exercises 368 - 381

Hints and Answers 368 - 381

B. Inverse Circular and Trigonometric Functions

Exercises 382 - 389

Hints and Answers 382 - 389

C. Exponential and Logarithmic Functions

Exercises 390 - 400

Hints and Answers 390 - 400

D. Hyperbolic and Inverse hyperbolic Functions

Exercises 401 - 408

Hints and Answers 401 - 408

E. Compound Functions

Exercises 409 - 454

Hints and Answers 409 - 454

F. Miscellaneous Functions

Exercises 455 - 563

554. Find f+(0) and f_(0) of the functions:

555. Find f(0) + xf'(0) of f(x)=e-x.

556. Find f(3)+(x - 3)f '(3) of the function .

557. Given the functions f(x) = tan x and j(x) = ln (l - x), find f '(0)/f(0).

558. Given the functions f(x)= 1 - x and f(x)= 1 - sin p x/2, find f'(1)/f'(1).

559. Prove that the derivative of an even function is an odd function, the derivative of an odd function an even function.

560. Prove that the derivative of a periodic function is also a periodic function.

561. Show that the function y=xe-x satisfies the equation xy' = (1 - x)y.

562. Show that the function y=xe-x²/2 satisfies the equation xy' = (1 - x²)y.

563. Show that y = 1/(1 + x + ln x) satisfies the equation xy'=y(y ln x - 1).

Hints and Answers 455 - 563

G. Logarithmic Derivative

A logarithmic derivative of a function y =f(x) is the derivative of the logarithm of the function, i.e.,

Finding the derivative is sometimes simplified by first taking the logarithm of the function.

Example. Find the derivative of the exponential function

where u = j(x) and v = y(x).

Solution: Taking logarithms, we get

Differentiate both sides of this equation with respect to x:

whence

Exercises 564 - 580

564. Example. Find y', if

Solution:

565. Example. Find y', if y = (sin x)x.

Solution:

Hints and Answers 566 - 580

2.3 Derivatives of Functions Not Represented Explicitly

2.3.1 Derivative of an inverse function: If a function y = f(x) has a derivative yx' ¹ 0, then the derivative of the inverse function x = f -1(y) is

Example 1. Find the derivative x'y if

Solution: We have

2.3.2 The derivatives of parametrically represented functions: If a function y is related to an argument x by means of a parameter t

then

Example 2. Find dy/dx if

Solution: We find

2.3.3 Derivative of an implicit function: If the relationship between x and y is given implicitly

then, in order to find the derivative y'x = y' in the simplest cases, it is sufficient

1) to calculate the derivative, with respect to x, of the left hand side of (1), taking y as a function of x,
2) to equate this derivative to zero, i.e., to set

3) to solve the resulting equation for y.

Example 3. Find the derivative y' if

Solution: Forming the derivative of the left hand side of (3) and equating it to zero, we get

Exercises 581 - 620

581. Find the derivative x'y if

In the following problems, find the derivative y' = dy/dx of the functions y represented parametrically:

595. Calculate dy/dx when t = p/2 if

Solution:

.

In the following examples, one has to find the derivative y' = dy/dx of implicit functions y.

Hints and Answers 582 - 620

2.4 Geometry and Mechanics Applications of the Derivative

2.4.1 Equations of the tangent and the normal: It follows from the geometric significance of the derivative that the equation of the tangent to a curve y = f(x) or F(x,y) = 0 at a point M(x0, y0) will be

where y0' is the value of the derivative y' at the point M(x0, y0). The straight line passing through the point where the tangent touches the curve, perpendicularly to the tangent, is called the normal to the curve. The normal has the equation

2.4.2 Angle between curves: The angle between the curves

at their common point M0(x0, y0) (Fig. 12) is the angle w between the tangents M0A and M0B to these curves at the point M0.

Using a familiar formula of analytic geometry, we find

 

2.4.3 Segments associated with the tangent and the normal in an orthogonal co-ordinate system: The tangent and the normal determine the four segments:

Since KM = |y0| and tan j = y0', It follows that

2.4.4 Segments associated with the tangent and the normal in a polar co-ordinate system: If a curve is given in polar co-ordinates by the equation r =f(j), then the angle m formed by the tangent MT and the radius vector r = OM (Fig.14) is defined by

The tangent MT and the normal MN at the point M together with the radius s vector of the point of tangency and with the perpendicular to the radius vector drawn through the pole O determine the following four segments (Fig. 14):

These segments are given by the formulae:

Exercises 621 - 666

621: What angles j are formed with the x-axis by the tangents to the curve y = x - x² at the points with the abscise:

Solution:. We have y' = l - 2x, whence

622: At what angles do the curves y= sin x and y = sin 2x intersect the abscissae at the origin?

623: At what angle does y = tan x intersect the abscissa at the origin?

624: At what angle does the curve y=e0.5x intersect the straight line x=2?

625. Find the points at which the tangents to the curve

are parallel to the x-axis.

626. At what point are the tangent to the parabola y = x² - 7x = 3 and the straight line 5x + y - 3 = 0 parallel?

627. Find the equation of the parabola y = x² + bx + c which is tangent to the straight line x = y at the point (1,1).

628. Determine the slope of the tangent to the curve x³ + y³ - xy - 7 = 0 at the point (1, 2).

629. At what point of the curve y² = 2x³ is the tangent perpendicular to the straight line 4x - 3y + 2 = 0?

630. Write the equation of the tangent and the normal to the parabola

at the point with abscissa x = 4.

Solution: We have

whence the slope of the tangent is k = |y'|x=4 = 1/4. Since the point of the tangent has the co-ordinates x = 4, y = 2, the equation of the tangent is y - 2 = l/4(x - 4) or x - 4y + 4 = 0.

Since the slope of the normal must be perpendicular,

whence the equation of the normal is

631. Write the equations of the tangent and the normal to the curve y = x³ + 2x² - 4x - 3 at the point (-2,5).

632. Find the equations of the tangent and .the normal to the curve

at the point (1,0).

633. Form the equations of the tangent and the normal to the curves at the given points:

a) y = tan 2x at the origin;
b) y = arsin[(x - 1)/2] at the intersection with the x-axis;
c) y = arcos 3x at the intersection with the y-axis;
d) y = ln x at the intersection with the .x-axis;
e) y = e1-x² at the intersection with the straight line y = 1

634. Write down the equations of the tangent and the normal to the curve

at the point (2,2) .

635. Find the equations of the tangent to the curve

at the origin and the point t = p/4.

636. Find the equations of the tangent and the normal to the curve

at the point with ordinate y = 3.

637. Find the equation of the tangent to the curve x5 + y5 - 2xy = 0 at the point (1,1).

638. Find the equations of. the tangents and normals to the curve y = (x - 1)(x - 2) (x - 3) at its intersection with the x-axis.

639. Find the equations of the tangent and the normal to the curve y4 = 4x4 + 6xy at the point (1,2).

640*. Show that the segment of the tangent to the hyperbola xy = a² (the segment lies between the co-ordinate axes) is divided in two at the point of tangency.

641. Show that in the case of the astroid x2/3 + y2/3 = a2/3 the segment of the tangent between the co-ordinate axes has the .constant value a.

642. Show that the normals to the involute of the circleare tangents to the circle

643. Find the angle at which the parabolas y= (x - 2)² and y = -4 + 6x - x² intersect.

644. At what angle do the parabolas y=x² and y = x³ intersect?

645. Show that the curves y = 4x² +2x -8 and y = x³ - x + 10 are tangent to each other at the point (3, 34). Will we have the same thing at (-2, 4)?

646. Show that the hyperbolas xy = a², x² - y = b² intersect at a right angle.

647. Given a parabola y² = 4x, evaluate at the point (1,2) the lengths of the segments of the sub-tangent, sub-normal, tangent and normal.

648. Find the length of the segment of the sub-tangent of the curve y = 2x at any of its points.

649. Show that in the equi-lateral hyperbola x² - y² = a² the length of the normal at any point is equal to the radius vector of that point.

650. Show that the length of the segment of the subnormal in the hyperbola x² - y² = a² at any point is equal to the abscissa of this point.

651. Show that the segments of the sub-tangents of the ellipse x²/a² - y²/b² = 1 and the circle x² + y² = a² at points with the same abscissas are equal. What procedure of construction of the tangent to the ellipse follows from this?

652. Find the length of the segment of the tangent, the normal, the sub-tangent and the sub-normal of the cycloid

at an arbitrary point t = t0.

653. Find the angle between the tangent and the radius vector of the point of tangency in the case of the logarithmic spiral

654. Find the angle between the tangent and the radius vector of the point of tangency for the lemniscate

655. Find the lengths of the segments of the polar sub-tangent, sub-normal, tangent and normal as well as the angle between the tangent and the radius vector of the point of tangency in the case of the spiral of Archimedes

at the point with the polar angle j = 2p.

656. Find the lengths of the segments of the polar sub-tangent, sub-normal, tangent and normal as well as the angle between the tangent and the radius vector of the hyperbolic spiral r = a/j at an arbitrary point j = j0, r = r0+.

657. The law of motion of a point on the x-axis is

Find the velocity of the point at t0 = 0, t1= 1, t2 =2 (x is in centimetres and t in seconds).

658. Two points move along the x-axis with the laws of motion

where t ³ 0. At what speed are these points receding from each other at the time of encounter (x is in centimetres, t is in seconds)?

659. The end-points of a segment AB = 5 m are sliding along t:he co-ordinate axes OX and OY (Fig. 16), A is moving at 2 m/sec. What is the velocity of B when A is at a distance OA = 3 m from the origin?

660*. The law of motion of a material point thrown up at an angle a to the horizontal with initial velocity v0 (in the vertical plane OXY in Fig. 17 is given by the formulae (air resistance being neglected):

where t is the time and g is the acceleration of gravity. Find the trajectory of motion and the distance covered. Moreover, determine its velocity and its direction of motion.

661. A point is in motion along the hyperbola y = 10/x so that its abscissa x increases uniformly at a rate of 1 unit per second. What is the rate of change of its ordinate when the point passes through (5,2)?

662. At what point of the parabola y² = 18x does the ordinate increase at twice the rate of the abscissa?

663. One side of a rectangle, a = 10 cm, is of constant length, while the other side b increases at a constant rate of 4 cm/sec. At what rate are the diagonal of the rectangle and its area increasing when b = 30 cm?

664. The radius of a sphere is increasing at a uniform rate of 5 cm/sec. At what rate increase the. area of the surface and the volume of the sphere when the radius becomes 50 cm?

665. A point is in motion along the spiral of Archimedes

(a = 10 cm) so that the angular velocity of rotation of its radius vector is constant and equals 6° per second. Determine the rate of elongation of the radius vector r when r = 25 cm.

666. A non-homogeneous bar AB is 12 cm long. The mass of a part of it, AM, increases with the square of the distance of the moving point M from the end A and is 10 gm when AM =2 cm. Find the mass of the entire .bar AB and the linear density at any point M. What is the linear density of the bar at A and B?

Hints and Answers 621 - 666

2.5 Derivatives of Higher Orders 


2.5.1 Definition of higher order derivatives: A derivative of the second order or the second derivative of the function y=f(x) is the derivative of its derivative, i.e.,

The second derivative may be represented by

If x = f(t) is the law of rectilinear motion of a point, then d²x/dt² is the acceleration of this motion.

In general, the n-th derivative of a function y = f(x) is the derivative of a derivative of order (n - 1). We use for the notation of the n-th derivative

Example 1. Find the second derivative of the function

Solution:

2.5.2 Leibnitz's rule: If the functions u = j(x) and v = y(x) have derivatives up to the n-th order inclusively, then we can use the Leibnitz's rule (or formula) to evaluate the n-th derivative of a product of these functions:

2.5.3 Higher-order derivatives of parametrically represented functions:  If

then the derivatives y'x = dy/dx, y"xx, ··· can successively be calculated by means of the formulae

For a second derivative, we have the formula

Example 2. Find y", if

Solution: We have

Exercises 667 - 691

A. Higher-order derivatives of explicit functions

Find the second derivatives of the functions:

Hints and Answers 667 - 691

B. Higher-Order Derivatives of 
Parametrically Represented Functions and implicit Functions

Exercises 692 - 711

Find dy2/dx2 in the following problems:

Hints and Answers 692 - 698

Exercises 699 - 711

In the following examples, find y''' = d³y/dx³:

Hints and Answers 699 - 711

2.6 Differentials of First and Higher Orders

2.6.1 First-order differential: The differential (first-order) of a function is the principal part of its increment, which part is linear relative to the increment Dx = dx of the independent variable x. The differential of a function is equal to the product of its derivative and the differential of the independent variable

whence

If MN is an arc of the graph of the function y = f(x) (Fig. 19), MT is the tangent at M(x, y) and

then the increment in the ordinate of the tangent

and the segment AN = Dy.

Example 1. Find the increment and the differential of the function y = 3x² - x.

First solution:

whence

Second solution:

Example 2: Calculate Dy and dy of the function y = 3x² - x for x= l and Dx = 0.01.

Solution:

2.6.2 Principal properties of differentials:  

2.6.3. Applying differentials to approximate calculations: If the increment Dx of the argument x is small in absolute value, then the differential dy and the increment Dy of the function y = f(x) are approximately the same:

Example 3. By (approximately) how much does the side of a square change if its area increases from 9² m* to 9.1 m² ?

Solution: If x is the area of the square and y is its side, then

We know that x = 9 and Dx = 0.1. The increment Dy of the side of the square may be calculated approximately as follows:

2.6.4 Higher order differentials: A second-order differential is the differential of a first-order differential:  

We define similarly the differentials of the third and higher orders. If y = f(x) and x is the independent variable, then

However, if y = f(u), where u = j(x), then

(Here the primes denote derivatives with respect to u).

Exercises 712 - 755

712. Find the increment Dy and the differential dy of the function u = 5x + x² for x = 2 and Dx = 0.001.

713. Without calculating the derivative, find

714. The area of a square S with side x is given by S = x²'. Find the increment and the differential of this function and explain the geometric significance of the latter.

715. Give a geometric interpretation of the increment and differential of the functions:

a) the area of a circle, S = px²,
b) the volume of a cube, v =x³.

716. Show that when Dx ® 0, the increment in the function y = 2x, corresponding to an increment Dx in x, is for any x equivalent to the 2x ln 2 Dx.

717. For what value of x is the differential of the function y =x² not equivalent to the increment in this function as Dx®0?

718. Has the function y = |x| a differential for x = 0?

719. Using the derivative, find the differential of the function y= cos x for x = p/6 and Dx = p/36.

720. Find the differential of the function

for x = 9 and Dx = - 0.01.

721. Calculate the differential of the function

for x = p/3 and Dx = p/180.

In the following problems, find the differentials of the given functions for arbitrary values of the argument and its increment:

731. Find dy if x² + 2xy - y² = a².

Solution: Taking advantage of the invariance of the form of a differential, we obtain 2xdx + 2(ydx + xdy) = 2ydy = 0, whence

In the following examples find the differentials of functions defined implicitly.

735. Find dy at the point (1,2), if y³ - y = 6x².

736. Find the approximate value of sin 31°.

Solution: Setting x = arc 30º = p /6 and Dx = arc 1º = p /180º, we have sin 31º » sin 30º + p/180 cos 30º = 0.500+0.017À“3/2 = 0.015.

737. Replacing the increment of the function by the differential, calculate approximately:

738. What will be the approximate increase in the volume of a sphere if its radius R = 15 cm increases by 2 mm?

739. Derive the approximate formula, for |Dx| which are small compared with x,

Using it, approximate

740. Derive the approximate formula

and find approximate values of

741. Approximate the functions:

742. Approximate tan 45° 3' 20".

743. Find the approximate value of arsin 0.54.

744. Approximate

745. Using Ohm's law I = E/R, show that a small change in the current, due to a small change in the resistance, may be found approximately by the formula

746. Show that, in determining the length of the radius, a relative error of l% results in a relative error of approximately 2% in a calculation of the area of a circle and the surface of a sphere.

747. Compute d²y, if y = cos 5x.

Solution:

Hints and Answers 712 - 755

2.7 Mean value theorems

2.7.1 Rolle's theorem:   If a function f(x) is continuous on the interval a £ x £ b, has a derivative f '(x) at every interior point of this interval and

then the argument x has at least one value x, where a < x < b such that

2.7.2 Lagrange's theorem:   If a function f(x) is continuous on the interval a < x < b and has a derivative at every interior point of this interval, then

where a < x < b.

2.7.3 Cauchy's theorem:  If the functions f(x) and F(x) are continuous on the interval a < x < b, have there derivatives which do not vanish simultaneously and F(b) ¹ F(a), then

Exercises 756 - 765

756. Show that the function f(x) = x - x² satisfies on the intervals -l £ x £ 0 and 0 £ x £ l Rolle's theorem. Find the values of x.

Solution: The function f(x) is continuous and differentiable for all values of x and f(-l) = f(0) = f(l) = 0. Hence, Rolle's theorem is applicable on the intervals -1 £ x £ 0 and 0 £ x £ l. In order to find x, we form the equation

whence

where -1 < x1 < 0 and 0 < x2 < 1.

757. The function

has equal values

at the end-points of the interval [0,4]. Does Rolle's theorem hold for this function on [0,4]?

758. Does Rolle's theorem hold for the function

on the interval [0, p)?

759. Let

Show that the equation has three real roots.

760. Obviously, the equation

f '(x) = 0 has a root x = 0. Show that this equation cannot have any other real root.

761. Test whether Lagrange's theorem holds for the function

in the interval [-2, 1) and find the appropriate intermediate value of x.

Solution: The function is continuous and differentiable for all values of x and f '(x) = l - 3x², whence, by Lagrange's formula, we have

Hence, 1 - 3x² = - 2 and x = ± l; the only suitable value is x = -1, for which -2 < x < 1.

762. Test the validity of Lagrange's theorem and find the appropriate intermediate point x for the function f(x) = x4/3 in the interval [-1, 1].

763. Given a segment of the parabola y = x² lying between two points A(l, l) and B(3, 9), find a point at which the tangent is parallel to the chord AB.

764. Using Lagrange's theorem, prove the formula

where x < x < x + h.

765. a) For the functions f(x) = x² + 2 and F(x) = x³ - 1, test whether Cauchy's theorem holds on the interval [1, 2] and find x;
b) Repeat for f(x) = sin x and F(x) = cos x on the interval [0, p/2].

Hints and Answers 757 - 765

2.8. Taylor's Formula  

If a function f(x) is continuous and has continuous derivatives up to the (n - l)-th order inclusively in the interval 0 £ :x £ b (or b £ x £ a), and there is a finite derivative f (n)(x) at each interior point of the interval, then Taylor's formula

where x = a + q (x - a) and 0 < q < l, holds true in the interval.

In particular, when a = 0, one has Maclaurin's formula

where x = q x, 0 < q < 1.

Exercises 766 - 775

766: Expand the polynomial

in positive integral powers of x - 2.

Solution:

for n = 4, whence

Thus,

or

767. Expand the function f(x) = ex in powers of x + 1 to the term containing (x + 1)³.

Solution: f'(n)(x) = ex for all n, f (n)(-l)= 1/e, whence

where x = - l + q (x + 1), 0 < q < l.

768. Expand the function f(x) = ln (x) in powers of x - 1 up to the term with (x - l)².

769. Expand f(x) = sin x in powers of x up to the term containing x3 and to the term containing x5.

770. Expand f(x) = ex in powers of x up to the term containing xn-1.

771. Show that sin (a + h) differs from

by not more than 1/2 h².

772. Determine the origin of the approximate formulae:

and evaluate their errors.

773. Evaluate the error in the formula

774. Due to its own weight, a heavy suspended thread lies in a catenary y = a cosh x/a. Show that for small |x| the shape of the thread is approximately expressed by the parabola

775*. Show that for to within (x/a)², we have approximately

Hints and Answers 766 - 775

2.9 L'Hospital - Bernoulli Rule for Evaluating Indeterminate Expressions

2.9.1 Evaluation of the indeterminate forms 0/0 and ¥/¥: Let the single-valued functions f(x) and j(x) be differentiable for 0 < |x - a| < h and the derivative of one of them not vanish.

If both f(x) and j(x) are infinitesimal or infinite as x ® a, i.e., if the quotient f(x)/j(x) at x=a has one of the indeterminate forms 0/0 or ¥/¥, then

provided that the limit of the ratio of the derivatives exists.

This rule is also applicable when a = ¥.

If the quotient f '(x)/j '(x) yields again at the point x = a an indeterminate form of one of the two above-mentioned types and f '(x) and j '(x) satisfy all the requirements stated above for f(x) and j(x), we can pass to the ratio of second derivatives, etc.

However, note that the limit of the ratio f(x)/j(x) may exist, whereas the ratios of the derivatives do not tend to any limit (Example 809).

2.9.2 Other Indeterminate forms: In order to evaluate an indeterminate form like 0·¥, transform the appropriate product f1(x)f2(x), where

into the quotient

In the case of the indeterminate form ¥ - ¥, one should transform the appropriate difference f1(x) - f2(x) into the product

and first evaluate the indeterminate form

if its limit as x ® a is 1, we reduce the expression to

 

The indeterminate forms

are evaluated by first taking logarithms and then finding the limit of the logarithm of the power

(which requires evaluating a form like 0·¥.)

In certain cases, it is useful to combine L'Hospital's rule with finding limits by elementary techniques.

Example 1 . Compute

Solution: Applying L'Hospital's rule, we have

We have the indeterminate form 0/0. However, we do not need to use L'Hospital's rule, since we know that

Thus, finally, we find

Example 2. Compute

Reducing this to a common denominator, we get

Before applying L'Hospital's rule, we replace the denominator of the last fraction by an equivalent infinitesimal (1.4)x2sin2x~x4. Thus, we obtain

L'Hospital's rule yields now

We find now by elementary means

Example 3: Compute

Taking logarithms and applying L'Hospital's rule, we get

whence

Exercises 776 - 810

Find the limits

776.

Solution:

 

Solution:

hints-and-answers 776 - 810

1 Chapter                            Chapter 3

2

Hit Counter