Front Matter1 Limits and Continuity2 Differentiation Rules3 Applications of Differentiation4 Parametric Equations and also Polar Coordinates6 Answers, Hints, Solutions7 Recommendations for Success in Mathematics
Authored in PreTeXt ## Chapter 5 True Or False and also Multiple Choice Problems

You are watching: Which of the following is false?

For each of the following ten statements answer TRUE or FALSE as appropriate:

If (f) is differentiable on (<-1,1>) then (f) is constant at (x=0 ext.)

If (f"(x)lt 0) and also (f"(x)>0) for all (x) then (f) is concave down.

The general antiderivative of (f(x)=3x^2) is (F(x)=x^3 ext.)

(ln x) exists for any kind of (x>1 ext.)

(ln x=pi) has actually a unique solution.

(e^-x) is negative for some values of (x ext.)

(ln e^x^2=x^2) for all (x ext.)

(f(x)=|x|) is differentiable for all (x ext.)

( an x) is defined for all (x ext.)

All crucial points of (f(x)) meet (f"(x)=0 ext.)

The feature (f(x)=left{ eginarraylll 3+fracsin (x-2)x-2amp mboxif amp x ot=2 \ 3amp mboxif amp x=2 endarray est.) is constant at all real numbers (x ext.)

If (f"(x)=g"(x)) for (0lt xlt 1 ext,) then (f(x)=g(x)) for (0lt xlt 1 ext.)

If (f) is enhancing and also (f(x)>0) on (I ext,) then (ds g(x)=frac1f(x)) is decreasing on (I ext.)

Tbelow exists a function (f) such that (f(1)=-2 ext,) (f(3)=0 ext,) and (f"(x)>1) for all (x ext.)

If (f) is differentiable, then (ds fracddxf(sqrtx)=fracf"(x)2sqrtx ext.)

(displaystyle ds fracddx10^x=x10^x-1)

Let (e=exp (1)) as usual. If (y=e^2) then (y"=2e ext.)

If (f(x)) and also (g(x)) are differentiable for all (x ext,) then (ds fracddxf(g(x))=f"(g(x))g"(x) ext.)

If (g(x)=x^5 ext,) then (ds lim _x o 2fracg(x)-g(2)x-2=80 ext.)

An equation of the tangent line to the parabola (y=x^2) at ((-2,4)) is (y-4=2x(x+2) ext.)

(displaystyle ds fracddx an ^2x=fracddxsec ^2x)

For all actual worths of (x) we have that (ds fracddx|x^2+x|=|2x+1| ext.)

If (f) is one-to-one then (ds f^-1(x)=frac1f(x) ext.)

If (x>0 ext,) then ((ln x)^6=6ln x ext.)

If (ds lim _x o 5f(x)=0) and also (ds lim _x o 5g(x)=0 ext,) then (ds lim _x o 5fracf(x)g(x)) does not exist.

If the line (x=1) is a vertical asymptote of (y=f(x) ext,) then (f) is not characterized at 1.

If (f"(c)) does not exist and also (f"(x)) alters from positive to negative as (x) increases via (c ext,) then (f(x)) has actually a regional minimum at (x=c ext.)

(sqrta^2=a) for all (a>0 ext.)

If (f(c)) exists but (f"(c)) does not exist, then (x=c) is a vital point of (f(x) ext.)

If (f"(c)) exists and also (f"""(c)>0 ext,) then (f(x)) has a neighborhood minimum at (x=c ext.)

Are the following statements TRUE or FALSE.

(ds lim _x o 3sqrtx-3=sqrtlim _x o 3(x-3) ext.)

(displaystyle ds fracddxleft( fracln 2^sqrtxsqrtx ight) =0)

If (f(x)=(1+x)(1+x^2)(1+x^3)(1+x^4) ext,) then (f"(0)=1 ext.)

If (y=f(x)=2^ ext,) then the variety of (f) is the set of all non-negative actual numbers.

(ds fracddxleft( fraclog x^2log x ight) =0 ext.)

If (f"(x)=-x^3) and also (f(4)=3 ext,) then (f(3)=2 ext.)

If (f"(c)) exists and if (f"(c)>0 ext,) then (f(x)) has a neighborhood minimum at (x=c ext.)

(ds fracdduleft( frac1csc u ight) =frac1sec u ext.)

(ds fracddx(sin ^-1(cos x)=-1) for (0lt xlt pi ext.)

(sinh ^2x-cosh ^2x=1 ext.)

(ds int fracdxx^2+1=ln (x^2+1)+C ext.)

(ds int fracdx3-2x=frac12ln |3-2x|+C ext.)

For all functions (f ext,) if (f) is constant at a particular allude (x_0 ext,) then (f) is differentiable at (x_0 ext.)

For all attributes (f ext,) if (ds lim _x o a^-f(x)) exist, and (ds lim _x o a^+f(x)) exist, then (f) is consistent at (a ext.)

For all functions (f ext,) if (alt b ext,) (f(a)lt 0 ext,) (f(b)>0 ext,) then tright here must be a number (c ext,) with (alt clt b) and (f(c)=0 ext.)

For all functions (f ext,) if (f"(x)) exists for all (x ext,) then (f"(x)) exists for all (x ext.)

It is difficult for a function to be discontinues at every number (x ext.)

If (f ext,) (g ext,) are any 2 features which are continuous for all (x ext,) then (ds fracfg) is consistent for all (x ext.)

It is feasible that features (f) and also (g) are not consistent at a suggest (x_0 ext,) however (f+g) is constant at (x_0 ext.)

If (ds lim _x o infty (f(x)+g(x))) exists, then (ds lim _x o infty f(x)) exists and (ds lim _x o infty g(x)) exists.

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(displaystyle ds lim _x o inftyfrac(1.00001)^xx^100000=0)

Every continuous attribute on the interval ((0,1)) has actually a maximum value and also a minimum value on ((0,1) ext.)

Answer each of the complying with either TRUE or FALSE.

Let (f) and also (g) be any 2 attributes which are consistent on (<0,1> ext,) through (f(0)=g(0)=0) and (f(1)=g(1)=10 ext.) Then tright here need to exist (c,din <0,1>) such that (f"(c)=g"(d) ext.)

Let (f) and also (g) be any 2 functions which are continuous on (<0,1>) and also differentiable on ((0,1) ext,) with (f(0)=g(0)=0) and also (f(1)=g(1)=10 ext.) Then there need to exist (cin <0,1>) such that (f"(c)=g"(c) ext.)