True and False (Please show solution or explanation if possible)

a. _____ ¹_-1 1/x²dx = [-1/x]¹_-1 = -2

b. _____If f is continuous then d/dx ^x_a f(t)dt= f(x).

c. _____ The slope of the tangent at any point (a, b) on the curve x² + y² = r² is -a/b.

d. ____ If f is continuous and, A(x) =^x_a f(t)dt, then A'(x) = f(x)

e. _____lim_x-0 x²/3x²+x = lim_x-0 2x/6x+1 = lim_x-0 2/6 =1/3

f. _____ The fact that f is an integrable function implies that there always exists a differentiable function, F(x), such that F'(x)= f(x).

g. _____lim_x-0 x²/3x²+x = lim_x-0 2x/6x+1 =1/3

h. _____ If f'(a) =g'(a) then f(a)=g(a) + c where c is a constant.

i. _____ If functions f, and g are differentiable, and have a maximum distance between the two functions at x=a, then f'(a)= g'(a).

j. _____ If f(x) is continuous on a closed interval I, and f(a) and f(b) have opposite signs where a and b are in I, then there exists a value c in [a, b] such that f (c) =0.

k. _____ There are only 2 types of asymptotes:horizontal, and vertical asymptotes.

l. ______ If a function is differentiable, then it must be continuous.

m. ____Since f(x)=1/x is continuous on (0, 1), f(x) is integrable on(0,1).

n. ______ lim_x-0 x² = 0.

o. ______ Every bounded continuous function is integrable.

p. ______ f(x)=|x| is not integrable in [-1, 1] because the function f is not differentiable at x=0.

q. ______If f(x) g(x) for every x in [a,b], then ^b_a f(x)dx < ^b_a g(x)dx , unless f(x)=g(x) for all x.

r. _______ You can always use a bisection algorithm to find a root of a continuous function.

s. ____ The L'Hospital's rule can be applied to find the limit of a function as long as you can express it into a ratio of two functions.

t. ____ ^b_a f(x) dx = (area above x- axis)- (area under x- axis)