# Integrals and continuity

20) If the function f is continuous for all real numbers and lim as h approaches 0 of f(a+h) - f(a)/ h = 7 then which statement is true?

a) f(a) = 7

b) f is differentiable at x=a.

c) f is differentiable for all real numbers.

d) f is increasing for x>0.

e) f is increasing for all real differentiable

ans is B. Explain

21) d/dx(sin(cosx)) = ? Explain

22) Which of the following is true for y=x^4-2x^3? Ans is the curve has two points of inflection and one relative extremum. Explain

23) Integral ofcos^2xsinxdx =? Explain

24) d/dxIntegral with x on top and 1 on bottom of t^2dt=? Explain

25) Let r(t) be a differentiable functon that is positive and increasing. The rate of increase of r^3 is equal to 12 times the rate of increase of r when r(t) = ?

Ans =2? Explain.

https://brainmass.com/math/integrals/integrals-continuity-calculus-15024

#### Solution Preview

20 B.

This is the definition of a function differentiable at a point x=a.

21 By chain rule, we have

d(sin(cosx))/dx=cos(cosx)*d(cosx)/dx=-sinx*cos(cosx)

22 y=x^4-2x^3

So we have y'=4x^3-6x^2, ...

#### Solution Summary

This looks at integrals and continuity of various functions.