(15) Evaluate each of the following integrals. (a) Z cos(x)
ln(sin(x)) dx (b) Z x arcsin(x 2 ) dx (c) Z 1 0 ln(1 + x 2 ) dx (d)
Z 1/4 0 arcsin(2x) dx
(16) Use the table of integrals to evaluate the integrals, if
needed. You may need to transform the integrand first.
(a) Z cos(4t) cos(5t)dt
(b) Z 1 cos3 (x) dx
(c) Z 1 x 2 + 6x + 9 dx
(d) Z 1 50 −...
Compute the Taylor series at x = 0 for ln(1+x) and for x cos x
by repeatedly differentiating the function. Find the radii of
convergence of the associated series.
Find dy/dx for a & b
a) sin x+cos y=1
b) cos x^2 = xe^y
c)Let f(x) = 5 /2 x^2 − e^x . Find the value of x for which the
second derivative f'' (x) equals zero.
d) For what value of the constant c is the function f continuous
on (−∞,∞)?
f(x) = {cx^2 + 2x, x < 2 ,
2x + 4, x ≥ 2}
Use two functions below for parts a and b.
?(?)=??−?
?(?)=ln(?)+ln(1−?)+3
a) Find the stationary points, if any, of the following functions
and label them accordingly (local or global minima/maxima or
inflection point).
b) Characterize the above functions as convex, concave or neither
convex nor concave
1. If f(x) = ln(x/4)
-(a) Compute Taylor series for f at c = 4
-(b) Use Taylor series truncated after n-th term to compute f(8/3)
for n = 1,.....5
-(c) Compare the values from above with the values of f(8/3) and
plot the errors as a function of n
-(d) Show that Taylor series for f(x) = ln(x/4) at c = 4 represents
the function f for x element [4,5]