Question

Can someone please do these questions step by step and explain them. I really need help so I can do the same for the rest.

a) f(x) = e^(sqrtx)
1. State the domain and range
2. Find f'(x)
3. Determine the intervals of increase and decrease.

d) f(x) = e^x + e^-x
1. State the domain and range
2. Find f'(x)
3. Determine the intervals of increase and decrease.

Answer 1

x must be positive if sqrt x is to be real.

then sqrt x is from 0 to + oo
e^0 = 1
so range would be from 1 to infinity

2. f' = e^sqrt x * d/dx (x^1/2)
=e^sqrt x * (1/2) x^-1/2
= 2 e^sqrt x / sqrt x

3. f' looks always positive to me

----------------------------
e^x + 1/e^x

think x = - some big number
say -100
then it is e^-100 + e^100
think x the same + big number
then it is e^100 + e^-100
the same y for + 100 and -100 and positive for both

now what if x is + or - 1
then it is e + 1/e around 3

now what if x = 0
1 + 1/1 = 2

so
domain is all real x
range is all y ≥ 2

I think you can do the rest now

Answer 2

a) y = e^(√x)

1. clearly x≥0 for the square root of x to be defined, so the domain is x ≥ 0
range : y ≥ 1
2. y' = (1/2)x^(-1/2)(e^(√x))
= 1/(2√x)(e^√x)
3. the graph is contantly increasing, never decreases


b) this is the equation of a "catenary", or the shape formed when a chain is allowed to hang freely.

it has a vertex when x = 0 giving f(0)= 2. Domain : x any real number, range : y ≥ 2

f'(x) = e^x - e^-x

for x < 0 the function is decreasing (the first derivative is negative)
and for x > 0 the function is increasing, because the derivative is positive.