Consider the function and the value of a. f(x) = 5 - x - x^2, a = 0
(a) Use Mtan = lim h->0 [f(a + h) – f(a)] / h
to find the slope of the tangent line mtan = f'(a).

Mtan=1-2a

Step-by-step explanation:

Mtan=lim h->0[5-(a+h)-(a+h)^2-(5-a-a^2)/h

=lim h->0[5-a-h-a^2-2ah-h^2-5+a+a^2]/h

=lim h->0[h-2ah-h^2+5-5-a+a-a^2+a^2]/h

=lim h->[h-2ah-h^2 +0+0+0]/h

=lim h->o[h(1-2a-h)/h

=lim h->0 (1-2a-h)

=1-2a-0

=1-2a

Mtan=1-2a.

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}