Recent questions tagged calculus

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91

Internet
For f(x)=√x x€[0,b] the number c satisfying mean value therom is c=1

For f(x)=√x x€[0,b] the number c satisfying mean value therom is c=1

249 views

Katyayani0306 asked Sep 21, 2022

0 votes

2 answers

92

TIFR CSE 2022 | Part A | Question: 6
Let $f$ be a polynomial of degree $n \geq 3$ all of whose roots are non-positive real numbers. Suppose that $f(1)=1$. What is the maximum possible value of $f^{\prime}(1)?$ $1$ $n$ $n+1$ $\frac{n(n+1)}{2}$ $f^{\prime}(1)$ can be arbitrarily large given only the constraints in the question

Let $f$ be a polynomial of degree $n \geq 3$ all of whose roots are non-positive real numbers. Suppose that $f(1)=1$. What is the maximum possible value of $f^{\prime}(1)...

724 views

admin asked Sep 1, 2022

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0 answers

93

TIFR CSE 2022 | Part A | Question: 14
Suppose $w(t)=4 e^{i t}, x(t)=3 e^{i(t+\pi / 3)}, y(t)=3 e^{i(t-\pi / 3)}$ and $z(t)=3 e^{i(t+\pi)}$ are points that move in the complex plane as the time $t$ varies in $(-\infty, \infty)$. Let $c(t)$ ... $\frac{1}{2 \pi}$ $2 \pi$ $\sqrt{3} \pi$ $\frac{1}{\sqrt{3} \pi}$ $1$

Suppose $w(t)=4 e^{i t}, x(t)=3 e^{i(t+\pi / 3)}, y(t)=3 e^{i(t-\pi / 3)}$ and $z(t)=3 e^{i(t+\pi)}$ are points that move in the complex plane as the time $t$ varies in $...

340 views

admin asked Sep 1, 2022

1 votes

2 answers

94

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 1
Evaluate the limit $ \lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x} $ $1$ $\frac{1}{2}$ $2$ $0$

Evaluate the limit$$\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}$$$1$$\frac{1}{2}$$2$$0$

767 views

GO Classes asked Aug 28, 2022

3 votes

1 answer

95

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 2
Determine the value of following limit $ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+4 x+1}-x\right) $ $2$ $4$ $\frac{1}{2}$ $3$

Determine the value of following limit$$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+4 x+1}-x\right)$$$2$$4$$\frac{1}{2}$$3$

640 views

GO Classes asked Aug 28, 2022

4 votes

3 answers

96

GO Classes Test Series 2023 | Calculus | Test 1 | Question: 3
The function $f(x)=x^{4}-6 x^{2}$ is increasing on the intervals $(0, \sqrt{3})$ only $(\sqrt{3}, \infty)$ only $(-\infty,-\sqrt{3})$ and $(0, \sqrt{3})$ only $(-\sqrt{3}, 0)$ and $(\sqrt{3}, \infty)$ only

The function $f(x)=x^{4}-6 x^{2}$ is increasing on the intervals$(0, \sqrt{3})$ only$(\sqrt{3}, \infty)$ only$(-\infty,-\sqrt{3})$ and $(0, \sqrt{3})$ only$(-\sqrt{3}, 0)...

666 views

GO Classes asked Aug 28, 2022

5 votes

1 answer

97

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 4
The function $f(x)=\cos x-x$ is an even function is an odd function is neither an even nor an odd function None of these

The function $f(x)=\cos x-x$is an even functionis an odd functionis neither an even nor an odd functionNone of these

518 views

GO Classes asked Aug 28, 2022

8 votes

2 answers

98

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 5
Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ $ \begin{aligned} &f(x)=1-x^{2 / 3}\\ &g(x)=x^{3}-2 x^{2}-x+2\\ &h(x)=\cos \left(\frac{\pi}{4}(x+1)\right) \end{aligned} $ Rolle's Theorem applies to: both $f$ and $g$ both $g$ and $h$ $g$ only $h$ only

Which of the following functions satisfy the conditions of Rolle's Theorem on the interval $[-1,1]?$ $$\begin{aligned}&f(x)=1-x^{2 / 3}\\&g(x)=x^{3}-2 x^{2}-x+2\\&h(x)=\c...

1.1k views

GO Classes asked Aug 28, 2022

4 votes

1 answer

99

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 6
Suppose that the derivative of a function $h$ is given by: $ h^{\prime}(x)=x(x-1)^{2}(x-2) $ On what interval(s) is $h$ increasing? $(-\infty, 0)$ $(-\infty, 0)$ and $(2, \infty)$ $(0,2)$ $(0,1)$ and $(2, \infty)$

Suppose that the derivative of a function $h$ is given by:$$h^{\prime}(x)=x(x-1)^{2}(x-2)$$On what interval(s) is $h$ increasing?$(-\infty, 0)$$(-\infty, 0)$ and $(2, \in...

632 views

GO Classes asked Aug 28, 2022

7 votes

3 answers

100

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 7
Let $q(x)$ be a continuous function which is defined for all real numbers. A portion of the graph of $q^{\prime}(x)$, the derivative of $q(x)$, is shown below. On which of the following interval(s) is $q(x)$ increasing? $(0,2)$ $(2,4)$ $(7,9)$ None of these

Let $q(x)$ be a continuous function which is defined for all real numbers. A portion of the graph of $q^{\prime}(x)$, the derivative of $q(x)$, is shown below.On which of...

932 views

GO Classes asked Aug 28, 2022

7 votes

2 answers

101

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 8
Choose the CORRECT statement - The function $f(x)=\exp \left(-x^{2}\right)-1$ has the root $x=0$. If a function $f$ is differentiable on $[-1,1]$, then there is a point $x$ in that interval where $f^{\prime}(x)=0$. If ... $f^{\prime \prime}(1)>0$ then there is a point in $(0,1)$, where $f$ has an inflection point.

Choose the CORRECT statement -The function $f(x)=\exp \left(-x^{2}\right)-1$ has the root $x=0$.If a function $f$ is differentiable on $[-1,1]$, then there is a point $x$...

1.1k views

GO Classes asked Aug 28, 2022

3 votes

1 answer

102

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 9
Evaluate $y^{\prime \prime}(1)$ where $y=e^{x}+x^{e}$. $0$ $1$ $e^{2}$ $e$

Evaluate $y^{\prime \prime}(1)$ where $y=e^{x}+x^{e}$.$0$$1$$e^{2}$$e$

478 views

GO Classes asked Aug 28, 2022

6 votes

1 answer

103

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 10
Consider the following statements: $f(x)$ is continuous on $[a, b]$ $f(x)$ is differentiable on $(a, b)$ $f(a)=f(b)$ Which of the above statements are required in order to guarantee a $c \in(a, b)$ such that $f^{\prime}(c)(b-a)=f(b)-f(a) ?$ I only I and II only I, II, and III I and III only

Consider the following statements:$f(x)$ is continuous on $[a, b]$$f(x)$ is differentiable on $(a, b)$$f(a)=f(b)$Which of the above statements are required in order to gu...

603 views

GO Classes asked Aug 28, 2022

7 votes

1 answer

104

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 11
Let $I=(a, b)$ be an open interval and let $f$ be a function which is differentiable on $I$. Which of the followings is/are true statements - If $f^{\prime}(x)=0$ for all $x \in I$, then there is a constant $r$ ... on $I$. If $f^{\prime}(x)>0$ for all $x \in I$, then $f(x)$ is strictly decreasing on $I$.

Let $I=(a, b)$ be an open interval and let $f$ be a function which is differentiable on $I$. Which of the followings is/are true statements -If $f^{\prime}(x)=0$ for all ...

622 views

GO Classes asked Aug 28, 2022

8 votes

1 answer

105

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 12
Which of the following is/are FALSE? The absolute maximum value of $f(x)=\dfrac{1}{x}$ on the interval $[2,4]$ is $2.$ If $f(x)$ is a continuous function and $f(3)=2$ and $f(5)=-1$, then $f(x)$ has a root between $3$ ... $h(1)=4$ and $h(2)=5$, then $h(x)$ has no roots between $1$ and $2.$

Which of the following is/are FALSE?The absolute maximum value of $f(x)=\dfrac{1}{x}$ on the interval $[2,4]$ is $2.$If $f(x)$ is a continuous function and $f(3)=2$ and $...

642 views

GO Classes asked Aug 28, 2022

6 votes

1 answer

106

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 13
Suppose $g(x)$ is a polynomial function such that $g(-1)=4$ and $g(2)=7$. Then there is a number $c$ between $-1$ and $2$ such that $g(c)=1$ $g^{\prime}(c)=1$ $g(c)=0$ $g^{\prime}(c)=0$

Suppose $g(x)$ is a polynomial function such that $g(-1)=4$ and $g(2)=7$. Then there is a number $c$ between $-1$ and $2$ such that$g(c)=1$$g^{\prime}(c)=1$$g(c)=0$$g^{\p...

729 views

GO Classes asked Aug 28, 2022

2 votes

2 answers

107

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 14
Which of the following limit is/are correct? $\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=1$ $\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=e$ $\displaystyle{}\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}=e^{2}$ $\displaystyle{}\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}=e$

Which of the following limit is/are correct?$\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=1$$\displaystyle{}\lim _{x \rightarrow \infty} \sqrt[x]{x}=e$$\displa...

527 views

GO Classes asked Aug 28, 2022

6 votes

1 answer

108

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 15
Suppose $f$ is twice differentiable with $ f^{\prime \prime}(x)=7 x-2, \quad f^{\prime}(-2)=0, \quad \text { and } \quad f(-2)=-2 . $ Find $f(0)$. $-337 / 6$ $-74 / 3$ $23 / 9$ $37 / 4$

Suppose $f$ is twice differentiable with$$f^{\prime \prime}(x)=7 x-2, \quad f^{\prime}(-2)=0, \quad \text { and } \quad f(-2)=-2 .$$Find $f(0)$.$-337 / 6$$-74 / 3$$23 / 9...

611 views

GO Classes asked Aug 28, 2022

4 votes

1 answer

109

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 16
The sum of three positive numbers is $12$ and two of them are equal. Find the largest possible product. $86$ $64$ $48$ $72$

The sum of three positive numbers is $12$ and two of them are equal. Find the largest possible product.$86$$64$$48$$72$

545 views

GO Classes asked Aug 28, 2022

3 votes

2 answers

110

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 17
If $f(x)=e^{x} g(x), g(0)=2$ and $g^{\prime}(0)=1$, then $f^{\prime}(0)$ is $1$ $3$ $2$ $0$

If $f(x)=e^{x} g(x), g(0)=2$ and $g^{\prime}(0)=1$, then $f^{\prime}(0)$ is$1$$3$$2$$0$

438 views

GO Classes asked Aug 28, 2022

6 votes

1 answer

111

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 18
Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime}(x) \geq 2$ for $x \in[1,6]$, then $f(6) \geq 8$ $f(6)<8$ $f(6)<5$ $f(6)=5$

Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime}(x) \geq 2$ for $x \in[1,6]$, then$f(6) \geq 8$$f(6)<8$$f(6)<5$$f(6)=5$

498 views

GO Classes asked Aug 28, 2022

8 votes

1 answer

112

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 19
The equation $x^{5}+x+1=0$ has a solution in the interval $[0,1]$ $[-1,0]$ $[-2,-1]$ $[1,2]$

The equation $x^{5}+x+1=0$ has a solution in the interval$[0,1]$$[-1,0]$$[-2,-1]$$[1,2]$

566 views

GO Classes asked Aug 28, 2022

2 votes

1 answer

113

GO Classes CS Test Series 2025 | Calculus | Topic Wise Test 3 | Question: 20
Which of the following expression evaluates to given integral $ \int \frac{\ln (\ln x)}{x \ln x} d x $ $\dfrac{\ln x}{x}+C$ $\frac{1}{2}(\ln \ln x)^{2}+C$ $(\ln x)^{2}+C$ $(\ln \ln x)+C$

Which of the following expression evaluates to given integral$$\int \frac{\ln (\ln x)}{x \ln x} d x$$$\dfrac{\ln x}{x}+C$$\frac{1}{2}(\ln \ln x)^{2}+C$$(\ln x)^{2}+C$$(\l...

590 views

GO Classes asked Aug 28, 2022

1 votes

0 answers

114

Engineering Mathematics
Please verify the solution(the differentiation part ),

Please verify the solution(the differentiation part ),

622 views

Overflow04 asked Aug 21, 2022

0 votes

1 answer

115

Maths for natural science
Determine the domain of the function $f(x)=\left | x \right |+1$

Determine the domain of the function $f(x)=\left | x \right |+1$

327 views

Hailemariam asked Aug 5, 2022

0 votes

0 answers

116

mathematics for natural science
Suppose a radioactive substance decays according to the equation $y=2000e^{-0.75t}$, where $y$ represent the mass in grams after $t$ hours. Assume that the half-life of radium is $1600$ years. If $y_{0}$ milligrams of radium are initially present, write the formula for the number of milligrams $y$ present after $t$ years.

Suppose a radioactive substance decays according to the equation $y=2000e^{-0.75t}$, where $y$ represent the mass in grams after $t$ hours.Assume that the half-life of r...

441 views

Hailemariam asked Aug 4, 2022

0 votes

1 answer

117

Mathematics for Natural Science
Determine the domain of the function $f(x) = |x – 2|$

Determine the domain of the function $f(x) = |x – 2|$

256 views

Hailemariam asked Aug 3, 2022

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