Test review please help

Question

Answer 1

14)

f(x) = -x³ + 7x + 1

Average rate of change of function in the interval [a,b] = [ f(b) - f(a) ] / [b-a]

f(4) = -4³ + 7(4) + 1 = - 64 + 28 + 1 = - 35

f(1) = -1³ + 7(1) + 1 = - 1 + 7 + 1 = 7

Average rate of change of function in the interval [1,4] = [ f(4) - f(1) ] / [4-1]

= [ - 35 - 7 ] / 3

= - 42 / 3

= - 14

Average rate of change of function is - 14.

Answer 2

16)

Given equation : 3x² - 2xy + y² = 11

Apply differentiation on both sides with respect to x.

3(2x) - 2 [ x(dy/dx) + y ] + 2y (dy/dx)  = 0

6x - 2x(dy/dx) - 2y + 2y (dy/dx)  = 0

6x - 2y   = 2x(dy/dx) - 2y (dy/dx)

[ 2x - 2y ] (dy/dx) = 2 ( 3x - y )

2(x - y) (dy/dx) = 2( 3x - y )

(dy/dx) = ( 3x - y ) /(x - y)

Substitute the point ( 1 , -2 ).

(dy/dx) = ( 3(1) - (-2) ) / (1 - (-2))

dy/dx = (3+2) /(1+2)

dy/dx = 5/3

The slope of the tangent line to the given curve is "5/3".

Answer 3

15)

Given equation : x³ + 2xy + y² = 1

Apply implicit differentiation on both sides with respect to x.

3x² + 2 [ x(dy/dx) + y ] + 2y (dy/dx)  = 0

3x² + 2x(dy/dx) + 2y + 2y (dy/dx)  = 0

3x² + 2y + (2x+2y)(dy/dx)  = 0

(2x+2y)(dy/dx)  = - 3x² - 2y

(dy/dx)  = (- 3x² - 2y) / (2x + 2y )

Substitute :  x = 1 , y = -2

dy/dx  =  ( - 3(1)² - 2(-2) ) / ( 2(1)+2(-2) )

dy/dx  =  ( - 3 + 4 ) / ( 2 - 4 )

dy/dx  =  ( 1 ) / ( - 2 )

dy/dx  =  -1/2

Solution : dy/dx  =  -½.