( ) ( )1 ( )1 ( ) ( ) ( )x ( )

Calculus I Practice problem set 9: Inverse Trigonometric Functions, Hyperbolic Functions.

1. Find the exact value of each expression.

(a) sin⁻¹

( )

1/2 (b) arctan

( )

−1 (c) ^sec−1

( )

²

(d) arcsin

( )

1 (e) tan⁻¹

(

tan3π 4

)

(f) cos

(

arcsin1 2

)

2. Find the derivative of the function.

(a) y= tan⁻¹x (b) h(x)= 1−x² arcsinx (c) f(x)= xln

(

arctanx

)

(d) h(t)=e^sec−1t (e) y= xcos⁻¹x− 1−x² (f) ^y=tan−1

(

^{x− 1+x2}

)

3. Evaluate the integral.

(a)

∫

¹ ₊

0

2 1

4 dt

t (b)

∫

₋ 2

4 1 t

dt (c)

∫

² ₊

0

cos2

1

π sin

xdx x

(d)

∫

^tan₁₊⁻_x^{12xdx (e)}

∫

₋ ^dx

x

x 4

1

2 (f)

∫

₋ ^dx

e e

x x

4 2

1 4. Find the numerical value of each expression.

(a) tanh0 (b) tanh1 (c) cosh3 (d) sinh1 (e) cosh(ln3) (f) sinh⁻¹1 5. Find the derivative.

(a) f(x)=tanh4x (b) g(x)=sinh² x (c) F(x)=sinhxtanhx (d) f(t)=e^tsech t (e) f(t)=ln(sinht) (f) y=sinh(coshx) 6. Evaluate the integral.

(a)

∫

^sinh

(

^1+4x

)

^{dx (b)}

∫

^{tanhxdx (c)}

∫

₂^sec_+tanh^h2x_x^dx

(d)

∫

₊

1 16t²

dt (e)

∫

₄₋¹_x^{2 dx (f)}

∫

₊ ^dx

x² 9 1

1

Derivatives of Inverse Trigonometric Functions

( )

2 1

1 sin 1

x dx x

d

= −

−

( )

2 1

1 cos 1

x dx x

d

− −

− =

(

¹

)

₂

1 tan 1

x x dx

d

= +

−

( )

1 csc 1

2 1

− −

=

−

x x dx x

d

( )

1 sec 1

2 1

= −

−

x x dx x

d

(

¹

)

₂

1 cot 1

x x dx

d

− +

=

−

Hyperbolic Functions:

sinh 2

x

x e

x e

− −

= ,

cosh 2

x

x e

x e

+ −

= Derivatives of Hyperbolic Functions

(

¹

)

₂

1 sinh 1

x dx x

d

= +

−

( )

1 cosh 1

2 1

= −

−

x dx x

d

(

¹

)

₂

1 tanh 1

x x dx

d

= −

−

( )

1 csch 1

2 1

− +

− =

x x dx x

d

( )

2 1

1 sech 1

x dx x

d

− −

− =

(

¹

)

₂

1 coth 1

x x dx

d

= −

−

Homework 9: Due next Wednesday before class time.

Problem 31,33,35,37,55,57,59,61 on page 492-493.

Problem 23,25,27,29,59,61,63,65 on page 484-485