Logarithmic

Question 1

Derive

e^x

Answer 1

e^x

Question 2

Analyze the following function

y = ln (x²-1)

Include: domain, intercepts, increase and decrease interval, P.O.I and curvature, and sketch.

Answer 2

Domain: x > 1, x<-1
Intercepts: no y, x = √2; x = -√2

Question 3

Write the 2 equations that describes e.

Answer 3

_x~>0lim (1+x)^1/x
_x~>∞lim (1+1/n)ⁿ

Question 4

Rewrite in term of e

_n~>∞lim (1+1/(n+100))ⁿ

Answer 4

e

Rewrite as p = x - 100

Question 5

Rewrite in term of e:

lim_x~>0(1+x/5)^1/x

Answer 5

e^1/5

Substitute x as 5t, and then rewrite as e.

Question 6

Derive

y = x²e^x2

Answer 6

2xe^x2(1+x²)

Question 7

Derive:

y = (e^x- e^-x)/(e^x+ e^-x)

Answer 7

(e^2x- 1)/(e^2x+ 1)

Question 8

Discuss

y = ln(x²- 1)

Answer 8

Domain: x∈(-∞,-1)U(1,∞)
Intercepts: no y-intercepts, roots: ±√2
Assymptote: VASM at ± 1
Increase: x∈(-∞,-1)
Decrease: x∈(1,∞)
Concave down: x∈(-∞,-1)U(1,∞)

Question 9

Derive

y = e^sin(x2)

Answer 9

2xe^sin(x2)cos(x²)

Question 10

Evaluate

lim_x~>0+ln(x)

Answer 10

-∞

Question 11

Evaluate

lim_x~>0-ln(x)

Answer 11

D.N.E

Question 12

Find value of K that creates real solution:

e^2x + ln K = 0

Answer 12

0 < K < e^9/4

Treat it like a quadratic equation and employs the determinant equation.

Question 13

Discuss:

y = ln(cos x)

Answer 13

Domain: (4n-1)π/2 < x < (4n+1)π/2; where n is real
Intercepts: y-intercepts: 0, roots: 2nπ
Assymptote: about the y-axis, period 2π
Increase: x∈((4n-1)π/2,2nπ)
Decrease: x∈(2nπ, (4n+1)π/2)
Max: (0, 2nπ)
Concave down: x∈((4n-1)π/2,(4n+1)π/2)

Examine only 1 intervals, and do it.

Question 14

Derive

y = x^x

Answer 14

x^x(lnx+1)

Question 15

Find dy/dx

y = (5x+1)⁶(5x³)^1/2(2x+27)^1/3/[(2x³+9)e^-2x+2(x+1)^1/5]

Answer 15

y[(30/5x-6)+(15x²/10x³+81)+(2/6x+81)-(6x²/2x³+9)+(1/5x+5)]