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# Integrals and continuity

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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.

##### Solution Summary

This looks at integrals and continuity of various functions.

##### 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, ...

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