Solve the below:
1 a) Find the following derivative implicitly with respect to x
If, Y=(1+xy)^(1/xy). Find dy/dx! without simplifying the derivative.
Compute dy/dx at (1, 1).
b) Find the formula for nth derivative of y=f(x)= √x
2a) let f(x)=(1+cos x) ^ sec(x)
find lim f(x)
x→ Π /2
b) y=(xy)^1/xy, find dy/dx and simlyfy y'( y prime)
evaluate y' at (e, 1/e)
3a) Find the equation of the linear aproximation of f(x)= sin(x). for x, near zero
b) y=f(x)= (ln x)^e^(1/cos(x)), f'(x)=dy/dx
4a)Use differental to aproximate cos(11/36 Π), hint f(x)=cos(x), choose x o= Π/3 (xo x of zero)
b) Evaluate √2, using Newton method for n=4 setps
5a) Let y=tan(x) and y=2x graph both equation and find the value of x where the two graph intersect, using Newtons method n= 4 steps use xo=1.25
b) Show that newtons method fails for y=x^1/3
xo=0.1
6a) The formula for amount A (maturity or future value in a savings account compounded n times a year for t years at interest rate of r and initial deposit p is given by the following fromula below A=p{1+ r/n)^nt, gind the formula for A if the number of compounding periood per year becomes larger or n→∞
b)Find the derivative of the new ormula in 6a) with respect to t
7a) lim (1 - 2/x)^x
x→∞
b) lim (sin)^x
x→o+
8a)If f(x)=x^3 +x-1 use, rolles theorem and intermediate value theorem to show that f has exactly one real root
b)Find the number c for f(X)=x-x on [0 2] where the instatenious rate of c is equal to the averge rate of change of f(X) on [0 2]
9a) Graph y=f(x)= x^x and find two critical numbers for f(x) and determine if f is differentable and continious on both numbers you have found
b)Determine the region in which f is concave up or down
10a) Let f(x)= 3x^4-16x^3+18x^2. find all critical numbers, the interval of increasing, decreasing, concavity, inflection point for problems
b) g(x)=x^? (6 - x)^?