Question 11 Mark
AnswerThe change in cost due to small change in production is called marginal cost.
View full question & answer→Question 21 Mark
What is marginal revenue?
AnswerThe change in revenue due to small change in demand is called marginal revenue.
View full question & answer→Question 31 Mark
What are the stationery points of a function?
AnswerThe points where maximum of minimum values occur are known as stationary points. The necessary condition to obtain a stationary value is $f'(x) = \frac{dy}{dx} = 0.$
View full question & answer→Question 41 Mark
How will be the second order derivative of a function at $x=a$ if function is maximum at $x = a?$
AnswerIf the function is maximum at $x = a$ the second order derivative of a function at $x = a$ will be $f"(a) < 0.$ It means negative.
View full question & answer→Question 51 Mark
How will be the first order derivation of a function at $x= a$ if function is decreasing at $x=a?$
AnswerIf the function is decreasing at $x = a$ the first order derivative of a function at $x = a$ will be $f'(a) < 0.$ It means negative.
View full question & answer→Question 61 Mark
State the rule for derivative product of two function of $x.$
AnswerIf $u$ and $v$ are differentiable function of $x$ and if $y = u.v$ then $\frac{dy}{dx} = u(\frac{dv}{dx}) + v(\frac{du}{dx}).$
View full question & answer→Question 71 Mark
Find $\frac{d y}{d x}$ if $y=a^n$, $a$ is constant.
AnswerIf $y=a^n$; $a =$ constant, then $\frac{d y}{d x} = 0.$
View full question & answer→Question 81 Mark
Find $f’(x)$ for the function $f’(x) = 50.$
AnswerConstant number is always $'0'.$
So, if function $f(x) = 50,$ then $f'(x) = 0.$
View full question & answer→Question 91 Mark
Find $\frac{d y}{d x}$ if $y=6 x^3+\frac{7}{2}+x^2+\frac{6}{5} x-8$.
Answer
| Here, $y = 6x^3 + \frac{7}{2} + x^2 + \frac{6}{5}x - 8$ |
| $\frac{dy}{dx}$ |
$=\frac{\mathrm{d}}{\mathrm{dx}}\left(6 x^3\right)+\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{7}{2} \mathrm{x}^2\right)+\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{6}{5} \mathrm{x}\right)-\frac{\mathrm{d}}{\mathrm{dx}}(8)$ |
| |
$=6 \frac{\mathrm{dy}}{\mathrm{dx}}\left(x^3\right)+\frac{7}{2} \frac{\mathrm{d}}{\mathrm{dx}}\left(x^2\right)+\frac{6}{5} \frac{\mathrm{d}}{\mathrm{dx}}(x)-\frac{\mathrm{d}}{\mathrm{dx}}(8)$ |
| |
$=6\left(3 x^2\right)+\frac{7}{2}(2 x)+\frac{6}{5}(1)-0$ |
| |
$=18 x^2+7 x+\frac{6}{5}$ |
| $\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=18 x^2+7 x+\frac{6}{5}$ |
View full question & answer→Question 101 Mark
Find $f^{\prime}(x)$ if $f(x)=7 x^2-6 x+5$.
AnswerIf $f(x)=7 x^2-6 x+5$ then $f^{\prime}(x)=14 x-6$.
View full question & answer→Question 111 Mark
State the formula of elasticity of demand.
AnswerElasticity of Demand = $\frac { Percentage\ Change\ in\ demand }{ Percentage\ Change\ in\ Price }$
If we denote the demand as $x$ and Price is $p$ and the demand function $x = f(p)$ is given then,
Elasticity of demand = -$\frac{p}{x} \frac{d x}{d p}$
View full question & answer→Question 121 Mark
AnswerLet $f : A \rightarrow R$ and $a ∈ A,$ where $A$ is and open interval of $R.$
If $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ exists, then this limit of a function $f$ is called derivative at $x = a.$
It is denoted by $f'(a).$
The process of obtaining derivative of function is called differentiation.
Thus, $f'(a) = \frac{f(a+h)-f(a)}{h}$
View full question & answer→Question 131 Mark
What are the conditions for revenue function $R$ to be maximum?
AnswerThe necessary and sufficient conditions for maximizing the revenue function $R$ are as follows:
$(i) \frac{dr}{dx} = 0$ and
$(ii) \frac{\mathrm{d}^2 \mathrm{R}}{\mathrm{dx}^2} = 0$
View full question & answer→Question 141 Mark
What is increment ratio of function $y=f(x) ?$
AnswerThe increment ratio of a function $y=f(x)$ is $\frac{\delta y}{\delta x}$
View full question & answer→Question 151 Mark
If $f(x)=a x+b$, find $f^{\prime \prime}(x)$
Answer$f(x)=a x+b: f^{\prime}(x)=a$ andf" $(x)=0$
View full question & answer→Question 161 Mark
If $x=$ demand and $p=$ price, then state the demand function.
AnswerIf $x=$ demand and $p=$ price, then demand function is $x=f(p)$.
View full question & answer→Question 171 Mark
If $y =(2 x+3)^{7}$, find $\frac{d y}{d x}$
View full question & answer→Question 181 Mark
If the second order derivative of a function at $x=a$ Is positive, then what will be the function at $x=a ?$
AnswerIf the second order derivative of a function at $x=a$ is positive, then the function will be minimum at $x=a$.
View full question & answer→Question 191 Mark
State the necessary condition for the stationary values of a function.
AnswerThe necessary condition for the stationary values of a function $y=f(x)$ is $\frac{d y}{d x}$ or $f^{\prime}(x)$ $=0$.
View full question & answer→Question 201 Mark
If the increase in the price of toys Is $25 \%$, then its demand Is decreased by $20 \%$. Find the elasticity of demand of toys.
View full question & answer→Question 211 Mark
$f: A \rightarrow R .$ State the symbol of derivative for a member $x$ of $A$ in domain of $f ?$
Answer$A \rightarrow R$, then the symbol of derivative for a member $x$ of $A$ in domain of $f$ is $f^{\prime}(x)$
View full question & answer→Question 221 Mark
If function $f: A \rightarrow R$, then what Is called the derivative of $f$ at a?
AnswerFor the function $f: A \rightarrow R$, if $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ exists, then this limit is called the derivative of $f$ at a.
View full question & answer→Question 231 Mark
Profit function Is a function of which variable?
AnswerProfit function is a function of production $($demand$) x$.
View full question & answer→Question 241 Mark
Why is the negative sign taken in the formula of elasticity of demand?
AnswerThe changes in the price and demand of an item are in opposite direction. Hence, the negative sign is taken in the formula of elasticity of demand.
View full question & answer→Question 251 Mark
By which derivative is marginal revenue or marginal cost obtained?
AnswerBy using the first order derivative marginal revenue or marginal cost is obtained.
View full question & answer→Question 261 Mark
State the sufficient condition for maximum profit $P.$
AnswerThe sufficient condition for maximum profit $P$ is $\frac{d^{2} P}{d x^{2}}<0( $Negative$).$
View full question & answer→Question 271 Mark
State the necessary condition for the stationary values of a function.
AnswerThe necessary condition for the stationary values of a function $y=f(x)$ is $\frac{d y}{d x}$ or $f^{\prime}(x)$ $=0$.
View full question & answer→Question 281 Mark
$y=f(x)$. If function is maximum at $x=a$, then for a small positive number $h$, what is the value of $f(a) ?$
Answer$y=f(x)$. If function is maximum at $x=a$, then for a small positive number $h, f(a)>f(a+h)$ and $f(a)$ $>f(a-h)$
View full question & answer→Question 291 Mark
State the sufficient condition for the function $f(x)$ to be minimum.
AnswerThe sufficient condition for the function $f(x)$ to be minimum is $f^{\prime \prime}(x)>0$, l.e., $f^{\prime \prime}(x)$ is positive.
View full question & answer→Question 301 Mark
If a function is increasing at $x$ what will be its first derivative?
AnswerIf a function is Increasing at $x$, then its first derivative will be positive, i.e., $f^{\prime}(x)>0 .$
View full question & answer→Question 311 Mark
If $f^{\prime \prime}(x)=\frac{x}{3}-1$ and $f^{\prime \prime}(2)=-\frac{1}{3}$, find $x$.
View full question & answer→Question 321 Mark
$f(x)=4 x-1$, find $f^{\prime \prime}(x)$
Answer$f(x)=4 x-1: f^{\prime}(x)=4$ and $f^{\prime \prime}(x)=0$
View full question & answer→Question 331 Mark
If $y =(2 x+3)^{7}$, find $\frac{d y}{d x}$
View full question & answer→Question 341 Mark
If $y=t$ and $t=\frac{3}{x}+1$, find $\frac{d y}{d x}$.
View full question & answer→Question 351 Mark
Write in words the rule of derivative of addition or subtraction of two functions of $x$.
AnswerThe derivative of addition or subtraction of two functions of $x$ is equal to the addition or subtraction of derivatives of the functions.
View full question & answer→Question 361 Mark
If $y =4 x$, find $\frac{d x}{d y}$
View full question & answer→Question 371 Mark
$y =\frac{t^{2}}{5} .$ Find $\frac{d y}{d t}$ and $\frac{d y}{d x} ?$
Answer$\frac{d y}{d t}=\frac{2 t}{5}$ and $\frac{d y}{d x}=0$
View full question & answer→Question 381 Mark
If $f(x)=x^{0}$, find $f^{\prime}(x)$
AnswerIf $f(x)=x^{0}=1$, then $f^{\prime}(x)=0$
View full question & answer→Question 391 Mark
In which problems is differentiation used?
AnswerDifferentiation is used in production, replacement pricing and other management decision problems.
View full question & answer→Question 401 Mark
When $\delta x \rightarrow 0$, what is the limit of $\frac{\delta y}{\delta x}$.
AnswerWhen $\delta x \rightarrow 0$, the limit of $\frac{\partial y}{\delta x}$ is $\frac{d y}{d x}$.
View full question & answer→Question 411 Mark
When $\delta x \rightarrow 0$, what is called the limit of $\frac{\delta y}{\delta x} \cdot ?$
AnswerWhen $\delta x \rightarrow 0$, then the limit of $\frac{\delta y}{\delta x}$ is called the derivative of $y$ with respect to $x$ and it is denoted by $\frac{\delta y}{\delta x}$.
View full question & answer→Question 421 Mark
What is increment ratio of function $y=f(x) ?$
AnswerThe increment ratio of a function $y=f(x)$ is $\frac{\delta y}{\delta x}$
View full question & answer→Question 431 Mark
$y f(x)$. By which symbols is the increase in $x$ and $y$ denoted?
Answer$y=f(x) .$ Then the increase in $x$ and $y$ is denoted by symbols $\delta x$ and $\delta y$ respectively.
View full question & answer→Question 441 Mark
How can we know how rapidly a function is changing at any point?
AnswerBy using differentiation we can know how rapidly a function is changing at any point.
View full question & answer→Question 451 Mark
Find $\frac {d y}{d x}$ if $y= 6 x^3+\frac 7 2 x^2+\frac 6 5 x-8.$
Answer$18 x^2+7 x+\frac{6} 5$
View full question & answer→Question 461 Mark
If $u$ and $v$ are differentiable functions of $x$ and $y = u v$, then rule for derivative for product of two functions of $x$ is as follows:
Answer$\frac{d y}{d x}=u \cdot \frac{d v}{d x}+v \cdot \frac{d u}{d x}$
View full question & answer→Question 471 Mark
How will be the second order derivative of a function at $x=a$ if function is minimum at $x=a?$
AnswerIf function is minimum at $x = a$ the second order derivative of a function at $x = a$ will be $f"(x) > 0.$
View full question & answer→Question 481 Mark
Find $\frac{\mathbf{d y}}{\mathbf{d x}}$ if $y=x^3+x^2+x-2$
AnswerHere, $y=x^3+x^2+x-2$
$\therefore \frac{d y}{d x} =\frac{d}{d x}\left( x ^3+ x ^2+ x -2\right)$
$=\frac{d}{d x}\left( x ^3\right)+\frac{d}{d x}\left( x ^2\right)+\frac{d}{d x}( x )-\frac{d}{d x}$
$=3 x^2+2 x+1-0$
$=3 x^2+2 x+1$
$(2)$
$\therefore \frac{d y}{d x}=3 x^2+2 x+11$
View full question & answer→Question 491 Mark
Why the negative sign is taken in the formulas of demand elasticity?
AnswerThe ratio of elasticity of demand is negative as the change in price and demand of a commodity is in opposite direction. For convenience, the value of elasticity of demand is taken positive and hence the negative sign is taken in the formul$(A).$
View full question & answer→Question 501 Mark
How the marginal revenue can be obtained?
AnswerMarginal revenue can be obtained by taking the derivative of revenue function with respect to $x.$ Thus, when the demand is $x$ then marginal revenue $= \frac{dr}{dx}.$
View full question & answer→