Ejercicios De Derivadas

Universidad Tecnológica del Valle de Toluca Carrera de Ingeniería Mecatrónica y Sistemas Productivos Cálculo 1, MT. 71 Profesor: M. en C. Erick Rojas Ramírez Part I.- Calculate thederivative in the follow expressions f (x) := (x2 + 3x8 )x2 (1 x)

Using the linear approximations, …nd the approximation at the point x0 = 5 and x0 = 5; the …gure 1 shows thebehaviour of f (x)

y

4e+6

2e+6

-5

-4

-3

-2

-1 -2e+6

1

2

3

4

x

5

-4e+6

-6e+6

…gure 1 Calculate the derivative in the follow expression f(x) := ln(x)2 + 1 and …nd the cuadratic approximation at the point x0 = 3 .Compare the approximation graphically with the …gure 2.

1

y4

3

2

1

-5

-4

-3

-2-1

1

2

3

4

x

5

Figure 2 Using the cuadratic approximation calculate: f (x) := cos( at the point x0 = 1:2 1 ) x2

y
0.8

0.6

0.4

0.2

0.0 0.8-0.2 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

x

Figure 3 2

Part II. In the next exercices compute the derivatives applying the chain rule theorem. f (x) f (x) f (x) = = = p7 x3 2 1 p yx 2 x3 + 2x5 + 3 (x3 + 2x5 + 3)1=4

The solution of y = sin2 (2x) is given by y 0 = 4 cos 2x sin 2x:Describe the procedure to obtain y 0 : and compare your result.Part IV. Calculate the follow exercices, please describe step by step the procedure for each exercice. y = sin(x) cos(x) + Solution: y 0 = cos2 x
1 tan2 x

1 tan(x)

tan2 x + 1sin2 x

y = tan2 (x) Solution: y 0 = 2 (tan x) tan2 x + 1 y = arccos(3x) + 1 x cos(x)

1 1 Solution: y 0 = x cos2 x sin x x2 cos x p1 39x2 Using the concept of derivative …nd thesolution of

y (3) y (4) y

1)3 + ln(x + 1) 1 = 3 sin( + ex+2 ) x = ln(cos(x) + 2x ) =

(x4

The deadline for this work is the next monday. Enjoy your homework !!!.

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