Differentiation

Mathematics 53 2nd Semester, A.Y. 2014-2015 Exercises 7 - Differentiation II Q3, R3, W8, X8 I. Apply the Chain Rule to find dy dx . There is NO need to simplify. 1. y = x -2/3 sec x + π 2 πx 2 - sin 3 (-5x) 2. y = sec 2 x +3x -2.5 6 √ x + 3 √ x + π 3. y = x +(x + sin 2 x) 3 4 4. y = sin 4 5 sin x cot 2x 5. y = cos 3 [ sec(tan(3x 4 + 5))] 6. y = r √ x cos x 3x 3 + tan 5x 7. y = cot - 3 x 3 + sin 4x x 3 csc x 8. y = cos 4 (3x) - 7 √ x tan(x 2 ) - 2e 9. y = csc 3 3x 2 + cot 5x πe + x +3 10. y = √ 3x 4 - 2 - π 2 cot(x 2 ) - 4 3 √ x II. Do as indicated. 1. Let f (x)= ( tan(1 - x), if x> 1 x 2 - x, if x ≤ 1 . Determine if f is differentiable at x = 1. 2. Let f (x)= ( x 2 + (1 - x) 4/3 , if x< 1 2x 2 - 2x, if x ≥ 1 . Determine if f is differentiable at x = 1. 3. Let f (x)= - 4 1+ x +2, if x ≤ 1 2x 3 - 3x 2 + x, if x> 1 . Determine if f is differentiable at x = 1. 4. Find the value of b so that f (x)= -b 2 +4b 4x , if 0 <x<b 1 - x 4 , if b ≤ x is differentiable at x = b. 5. Let f (x)= ax 2 - 3, if x ≤ 2 b √ x +2+ 9x 2 , if x> 2 . Find the values of a and b so that f is differentiable at x = 2. Exercises from sample exams, books, and the internet rperez

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Exercises on differentiation

Transcript of Differentiation

Mathematics 53 2nd Semester, A.Y. 2014-2015Exercises 7 - Differentiation II Q3, R3, W8, X8

I. Apply the Chain Rule to finddy

dx. There is NO need to simplify.

1. y =x2/3 secx+ 2

x2 sin3(5x)

2. y =sec2 x+ 3x2.5

6x+ 3x+

3. y =[x+ (x+ sin2 x)3

]44. y = sin4

(5 sinx

cot 2x

)5. y = cos3[ sec(tan(3x4 + 5))]

6. y =

x cosx

3x3 + tan 5x

7. y = cot

3x3 + sin 4xx3 cscx

8. y =

cos4(3x) 7x

tan(x2) 2e

9. y = csc3(

3x2 + cot 5x

e+ x+ 3

)

10. y =

3x4 2 2

cot(x2) 43x

II. Do as indicated.

1. Let f(x) =

{tan(1 x), if x > 1

x2 x, if x 1 . Determine if f is differentiable at x = 1.

2. Let f(x) =

{x2 + (1 x)4/3, if x < 1

2x2 2x, if x 1 . Determine if f is differentiable at x = 1.

3. Let f(x) =

4

1 + x+ 2, if x 1

2x3 3x2 + x, if x > 1. Determine if f is differentiable at x = 1.

4. Find the value of b so that f(x) =

b2 + 4b

4x, if 0 < x < b

1 x4, if b x

is differentiable at x = b.

5. Let f(x) =

ax2 3, if x 2

bx+ 2 +

9x

2, if x > 2

. Find the values of a and b so that f is differentiable

at x = 2.

Exercises from sample exams, books, and the internet rperez