Second Fundamental Theorem of Calculus

Q.

The function $f:\left[0,3\right]\to \left[1,29\right]$ defined by $f\left(x\right)=2x^3-15x^2+36x+1$ is

1. One-One and Onto

2. Onto but not One-One

3. One-One but not Onto

4. neither One-One nor Onto

Q.

The function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ given by $f\left(x\right)=x^3-1$ is

1. a One-One function.

2. an Onto function.

3. a Bijection

4. neither One-One nor Onto

Q. If $\underset{x\to 1}{lim}\frac{x+x^2+x^3+...+x^n-n}{x-1}=820,(n\in N)$ then the value of $n$ is equal to

Q.

The function $f:\left[0,3\right]\to \left[1,29\right]$ defined by $f\left(x\right)=2x^3-15x^2+36x+1$ is

1. One-One and Onto

2. Onto but not One-One

3. One-One but not Onto

4. neither One-One nor Onto

Q.

The value (s) of${\int }_0^1\frac{x^4{\left(1-x\right)}^4}{1+x^2}dx$ is (are)

1. $\frac{22}{7}-\mathrm{\pi }$

2. $\frac{2}{105}$

3. $0$

4. $\frac{71}{15}-\frac{3\mathrm{\pi }}{2}$

Q.

What is the integration of $\log \left(\frac{1-x}{x}\right)$ with limit $0$ to $1$?

Q.

What is the opposite of integration in calculus?

Q. If $\underset{0}{\overset{1}{\int }}\frac{e^t}{1+t}dt=a,$ then $\underset{0}{\overset{1}{\int }}\frac{e^t}{(1+t{)}^2}$ is equal to

1. $a+1+\frac{e}{2}$
2. $a+1+\frac{e^2}{2}$
3. $a+1-\frac{e}{2}$
4. $a-1+\frac{e}{2}$

Q. Let $F:R\to R$ be a thrice differentiable function. Supose that $F\left(1\right)=0,F\left(3\right)=–4$ and $F^{\prime }\left(x\right)<0$ for all $x\in (1/2,3)$. Let $f\left(x\right)=xF\left(x\right)$ for all $x\in R$.

If $\underset{1}{\overset{3}{\int }}{x}^2{F}^{\prime }\left(x\right)dx=-12$ and $\underset{1}{\overset{3}{\int }}{x}^3{F}^{\prime \prime }\left(x\right)dx=40$, then the correct expression(s) is(are)

1. $9f^{\prime }\left(3\right)+f^{\prime }\left(1\right)–32=0$
2. $\underset{1}{\overset{3}{\int }}f\left(x\right)dx=12$
3. $9f^{\prime }\left(3\right)-f^{\prime }\left(1\right)+32=0$
4. $\underset{1}{\overset{3}{\int }}f\left(x\right)dx=-12$

Q.

How do you integrate two variables?

Q. The value of $\underset{n\to \infty }{lim}\prod _{r=2}^n\frac{r^3+1}{r^3-1}$ is

(Here, $\prod$ stands for the product.)

1. $\frac{3}{2}$
2. $\frac{2}{3}$
3. $\frac{3}{4}$
4. $\frac{4}{3}$

Q.

What is the fundamental concept in calculus?

Q.

What is the sum rule for derivatives?

Q. Let $F:[3,5]\to R$ be a twice differentiable function on (3, 5) such that $F\left(x\right)=e^{-x}\underset{3}{\overset{x}{\int }}(3t^2+2t+4F^{\prime }(t\left)\right)dt$. If $F^{\prime }\left(4\right)=\frac{\alpha e^{\beta }-224}{(e^{\beta }-4{)}^2}$, then $\alpha +\beta$ is equal to

Q. The value of $\frac{29{\int }_0^1{(1-x^4)}^{7}dx}{10{\int }_0^1{(1-x^4)}^{6}dx}$ is

1. $\frac{29}{28}$
2. $\frac{28}{29}$
3. $\frac{14}{5}$
4. $\frac{5}{14}$

Q. Let $I_n={\int }_0^{\infty }{e}^{-x}(sinx{)}^{n}dx,nϵN,n>1$ then $\frac{I_{2008}}{I_{2006}}$ equals

1. $\frac{2008\times 2007}{{2008}^2+1}$
2. $\frac{2006\times 2004}{{2008}^2-1}$
3. $\frac{2007\times 2006}{{2008}^2+1}$
4. $\frac{2008\times 2007}{{2008}^2-1}$

Q.

The function $f\left(x\right)=1–e^{–\frac{x^2}{2}}$

1. Decreasing for all $x$

2. Increasing for all $x$

3. Decreasing for $x<0$ and increasing for $x>0$

4. Increasing for $x<0$ and decreasing for $x>0$

Q. The maximum value of the function $f\left(x\right)={\int }_0^{1}t\sin (x+\pi t)dt,x\in R$ is -

1. $\frac{1}{{\pi }^2}\sqrt{{\pi }^2+4}$
2. $\frac{1}{\pi }\sqrt{{\pi }^2+4}$
3. $\sqrt{{\pi }^2+4}$
4. $\frac{1}{2{\pi }^2}\sqrt{{\pi }^2+4}$

Q. Given a function $g$ continuous on $R$ such that $\underset{0}{\overset{1}{\int }}g\left(t\right)dt=2$ and $g\left(1\right)=5.$ If $f\left(x\right)=\frac{1}{2}\underset{0}{\overset{x}{\int }}(x-t{)}^{2}g(t)dt,$ then the value of $\left(f^{\prime \prime \prime }\right(1)-f^{\prime \prime }(1\left)\right)$ is equal to

1. $0$
2. $3$
3. $5$
4. $7$

Q. The correct evaluation of ${\int }_0^{\pi }|si{n}^{4}x|dx$ is [MP PET 1993]

1. 
2. 
3. 
4. 

Q. Let $f:(0,\infty )\to (0,\infty )$ be a differentiable function satisfying, $x\underset{0}{\overset{x}{\int }}(1-t)f\left(t\right)dt=\underset{0}{\overset{x}{\int }}tf\left(t\right)dt\forall x\in R^{+}$ and $f\left(1\right)=1.$ Then $f\left(x\right)$ can be

1. $\frac{1}{x}{e}^{(1-\frac{1}{x})}$
2. $\frac{1}{x^2}{e}^{(1-\frac{1}{x})}$
3. $\frac{1}{x^2}{e}^{(1+\frac{1}{x})}$
4. $\frac{1}{x^3}{e}^{(1-\frac{1}{x})}$

Q.

The numbers P, Q and R for which the function
$f\left(x\right)=P{e}^{2x}+Q{e}^x+Rx$ satisfies the conditions
$f\left(0\right)=-1,f^{{}^{\prime }}\left(log2\right)=31$ and ${\int }_0^{log4}\left[f\right(x)-Rx]dx=\frac{39}{2}$ are
given by

1. 
   

2. 
   

3. 
   

4. 
   

Q. Let $f\left(x\right)$ be an even function such that $f\left(x\right)+f(x-3)=x(x-3)+1.$ Then $\underset{0}{\overset{3}{\int }}\frac{f\left(x\right)dx}{(x^2-3x+1)}$ is

1. $-\frac{2}{3}$
2. $-\frac{3}{2}$
3. $\frac{2}{3}$
4. $\frac{3}{2}$

Q.

Evaluate the following definite integrals as limit of sums.
${\int }_0^4(x+e^{2x})dx.$

Q.

Evaluate the following definite integrals as limit of sums.
${\int }_0^5(x+1)dx.$

Q.

How do you find the absolute maximum and minimum of a function?

Q.

Evaluate :${\int }_{-10}^{10}\log \left[\frac{\left(a+x\right)}{\left(a-x\right)}\right]dx$

1. $0$

2. $-2\log \left(a+10\right)$

3. $2\log \left[\left(a+x\right)/\left(a-x\right)\right]$

4. $2\log \left(a+x\right)$

Q. The value of is equal to ${\int }_3^6(\sqrt{x+\sqrt{12x-36}}+\sqrt{x-\sqrt{12x-36}})dx$is equal to

1. $6\sqrt{3}$
2. $4\sqrt{3}$
3. $12\sqrt{3}$
4. $2\sqrt{3}$

Q. ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}cosec2xdx=$

1. 

2. 
   

3. 

4. None of these

Q. Let $f:R\to R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x\in R$. Consider the following statements.
$I.$ $f$ is an odd function.
$II$. $f$ is an even function.
$III$. $f$ is differentiable everywhere.
Then

1. $I$ is true and $III$ is false
2. $II$ is true and $III$ is false
3. both $I$ and $III$ are true
4. both $II$ and $III$ are true