If a=3 and b=−2, find the values of:
Question: If $a=3$ and $b=-2$, find the values of: (i) $a^{a}+b^{b}$ (ii) $a^{b}+b^{a}$ (iii) $(a+b)^{a b}$ Solution: (i) $a^{a}+b^{b}$ Here, $a=3$ and $b=-2$. Put the values in the expression $a^{a}+b^{b}$. $3^{3}+(-2)^{-2}$ $=27+\frac{1}{(-2)^{2}}$ $=27+\frac{1}{4}$ $=\frac{108+1}{4}$ $=\frac{109}{4}$ (ii) $a^{b}+b^{a}$ Here, $a=3$ and $b=-2$ Put the values in the expression $a^{b}+b^{a}$. $3^{-2}+(-2)^{3}$ $=\left(\frac{1}{3}\right)^{2}+(-8)$ $=\frac{1}{9}-8$ $=\frac{1-72}{9}$ $=-\frac{71}{9}...
Read More →Find the derivative of the following functions:
Question: Find the derivative of the following functions: (i) sinxcosx (ii) secx (iii) 5 secx+ 4 cosx (iv) cosecx (v) 3cotx+ 5cosecx (vi) 5sinx 6cosx+ 7 (vii) 2tanx 7secx Solution: (i) Letf(x) = sinxcosx. Accordingly, from the first principle, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $=\lim _{h \rightarrow 0} \frac{\sin (x+h) \cos (x+h)-\sin x \cos x}{h}$ $=\lim _{h \rightarrow 0} \frac{1}{2 h}[2 \sin (x+h) \cos (x+h)-2 \sin x \cos x]$ $=\lim _{h \rightarrow 0} \frac{1}{2 h}...
Read More →Find the intervals in which the following functions are strictly increasing or decreasing:
Question: Find the intervals in which the following functions are strictly increasing or decreasing: (a) $x^{2}+2 x-5$ (b) $10-6 x-2 x^{2}$ (c) $-2 x^{3}-9 x^{2}-12 x+1$ (d) $6-9 x-x^{2}$ (e) $(x+1)^{3}(x-3)^{3}$ Solution: (a) We have, $f(x)=x^{2}+2 x-5$ $\therefore f^{\prime}(x)=2 x+2$ Now, $f^{\prime}(x)=0 \Rightarrow x=-1$ Point $x=-1$ divides the real line into two disjoint intervals i.e., $(-\infty,-1)$ and $(-1, \infty)$. In interval $(-\infty,-1), f^{\prime}(x)=2 x+20$ $\therefore f$ is s...
Read More →Simplify the following:
Question: Simplify the following: (i) $3\left(a^{4} b^{3}\right)^{10} \times 5\left(a^{2} b^{2}\right)^{3}$ (ii) $\left(2 x^{-2} y^{3}\right)^{3}$ (iii) $\frac{\left(4 \times 10^{7}\right)\left(6 \times 10^{-5}\right)}{8 \times 10^{4}}$ (iv) $\frac{4 a b^{2}\left(-5 a b^{3}\right)}{10 a^{2} b^{2}}$ (v) $\left(\frac{x^{2} y^{2}}{a^{2} b^{3}}\right)^{n}$ (vi) $\frac{\left(a^{3 n-9}\right)^{6}}{a^{2 n-4}}$ Solution: (i) $3\left(a^{4} b^{3}\right)^{10} \times 5\left(a^{2} b^{2}\right)^{3}$ $=3 \time...
Read More →Find the intervals in which the function
Question: Find the intervals in which the function $f$ given by $f(x)=2 x^{3}-3 x^{2}-36 x+7$ is (a) strictly increasing (b) strictly decreasing Solution: The given function is $f(x)=2 x^{3}-3 x^{2}-36 x+7$. $f^{\prime}(x)=6 x^{2}-6 x-36=6\left(x^{2}-x-6\right)=6(x+2)(x-3)$ $\therefore f^{\prime}(x)=0 \Rightarrow x=-2,3$ The points $x=-2$ and $x=3$ divide the real line into three disjoint intervals i.e., $(-\infty,-2),(-2,3)$, and $(3, \infty)$ In intervals $(-\infty,-2)$ and $(3, \infty), f^{\p...
Read More →Find the derivative of cos x from first principle.
Question: Find the derivative of cosxfrom first principle. Solution: Letf(x) = cosx. Accordingly, from the first principle, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $=\lim _{h \rightarrow 0} \frac{\cos (x+h)-\cos x}{h}$ $=\lim _{h \rightarrow 0}\left[\frac{\cos x \cos h-\sin x \sin h-\cos x}{h}\right]$ $=\lim _{h \rightarrow 0}\left[\frac{-\cos x(1-\cos h)-\sin x \sin h}{h}\right]$ $=\lim _{h \rightarrow 0}\left[\frac{-\cos x(1-\cos h)}{h}-\frac{\sin x \sin h}{h}\right]$ $=-...
Read More →Find the intervals in which the function
Question: Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is (a) strictly increasing (b) strictly decreasing Solution: The given function is $f(x)=2 x^{2}-3 x$. $f^{\prime}(x)=4 x-3$ $\therefore f^{\prime}(x)=0 \Rightarrow x=\frac{3}{4}$ Now, the point $\frac{3}{4}$ divides the real line into two disjoint intervals i.e., $\left(-\infty, \frac{3}{4}\right)$ and $\left(\frac{3}{4}, \infty\right)$. In interval $\left(-\infty, \frac{3}{4}\right), f^{\prime}(x)=4 x-30$. Hence...
Read More →Show that the function given by f(x) = sin x is
Question: Show that the function given byf(x) = sinxis (a) strictly increasing in $\left(0, \frac{\pi}{2}\right)$ (b) strictly decreasing in $\left(\frac{\pi}{2}, \pi\right)$ (c) neither increasing nor decreasing in $(0, \pi)$ Solution: The given function isf(x) = sinx. $\therefore f^{\prime}(x)=\cos x$ (a) Since for each $x \in\left(0, \frac{\pi}{2}\right), \cos x0$, we have $f^{\prime}(x)0$. Hence, $f$ is strictly increasing in $\left(0, \frac{\pi}{2}\right)$. (b) Since for each $x \in\left(\f...
Read More →Find the derivative of
Question: Find the derivative of (i) $2 x-\frac{3}{4}$ (ii) $\left(5 x^{3}+3 x-1\right)(x-1)$ (iii) $x^{-3}(5+3 x)$ (iv) $x^{5}\left(3-6 x^{-9}\right)$ (v) $x^{-4}\left(3-4 x^{-5}\right)$ (vi) $\frac{2}{x+1}-\frac{x^{2}}{3 x-1}$ Solution: (i) Let $f(x)=2 x-\frac{3}{4}$ $f^{\prime}(x)=\frac{d}{d x}\left(2 x-\frac{3}{4}\right)$ $=2 \frac{d}{d x}(x)-\frac{d}{d x}\left(\frac{3}{4}\right)$ $=2-0$ $=2$ (ii) Let $f(x)=\left(5 x^{3}+3 x-1\right)(x-1)$ By Leibnitz product rule, $f^{\prime}(x)=\left(5 x^{...
Read More →Show that the function given by
Question: Show that the function given byf(x) =e2xis strictly increasing onR. Solution: Let $x_{1}$ and $x_{2}$ be any two numbers in $\mathbf{R}$. Then, we have: $x_{1}x_{2} \Rightarrow 2 x_{1}2 x_{2} \Rightarrow e^{2 x_{1}}e^{2 x_{2}} \Rightarrow f\left(x_{1}\right)f\left(x_{2}\right)$ Hence,fis strictly increasing onR....
Read More →Show that the function given by
Question: Show that the function given by $f(x)=3 x+17$ is strictly increasing on $\mathbf{R}$. Solution: Letbe any two numbers inR. Then, we have: $x_{1}x_{2} \Rightarrow 3 x_{1}3 x_{2} \Rightarrow 3 x_{1}+173 x_{2}+17 \Rightarrow f\left(x_{1}\right)f\left(x_{2}\right)$ Hence,fis strictly increasing onR....
Read More →The total revenue in Rupees received from the sale of x units of a product is given by
Question: The total revenue in Rupees received from the sale ofxunits of a product is given by $R(x)=3 x^{2}+36 x+5$. The marginal revenue, when $x=15$ is (A) 116 (B) 96 (C) 90 (D) 126 Solution: Marginal revenue is the rate of change of total revenue with respect to the number of units sold. $\therefore$ Marginal Revenue $(\mathrm{MR})=\frac{d R}{d x}=3(2 x)+36=6 x+36$ $\therefore$ When $x=15$ $M R=6(15)+36=90+36=126$ Hence, the required marginal revenue is Rs 126. The correct answer is D....
Read More →Find the derivative offor some constant a.
Question: Find the derivative of $\frac{x^{n}-a^{n}}{x-a}$ for some constant $a$. Solution: Let $f(x)=\frac{x^{n}-a^{n}}{x-a}$ $\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(\frac{x^{n}-a^{n}}{x-a}\right)$ By quotient rule, $f^{\prime}(x)=\frac{(x-a) \frac{d}{d x}\left(x^{n}-a^{n}\right)-\left(x^{n}-a^{n}\right) \frac{d}{d x}(x-a)}{(x-a)^{2}}$ $=\frac{(x-a)\left(n x^{n-1}-0\right)-\left(x^{n}-a^{n}\right)}{(x-a)^{2}}$ $=\frac{n x^{n}-a n x^{n-1}-x^{n}+a^{n}}{(x-a)^{2}}$...
Read More →The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
Question: The rate of change of the area of a circle with respect to its radiusratr= 6 cm is (A) $10 \pi$ (B) $12 \pi$ (C) $8 \pi$ (D) $11 \pi$ Solution: The area of a circle(A)with radius (r) is given by, $A=\pi r^{2}$ Therefore, the rate of change of the area with respect to its radiusris $\frac{d A}{d r}=\frac{d}{d r}\left(\pi r^{2}\right)=2 \pi r$ $\therefore$ When $r=6 \mathrm{~cm}$, Hence, the required rate of change of the area of a circle is $12 \pi \mathrm{cm}^{2} / \mathrm{s}$. The cor...
Read More →For some constants a and b, find the derivative of
Question: For some constantsaandb, find the derivative of (i) $(x-a)(x-b)$ (ii) $\left(a x^{2}+b\right)^{2}$ (iii) $\frac{x-a}{x-b}$ Solution: (i) Let $f(x)=(x-a)(x-b)$ $\Rightarrow f(x)=x^{2}-(a+b) x+a b$ $\therefore f^{\prime}(x)=\frac{d}{d x}\left(x^{2}-(a+b) x+a b\right)$ $=\frac{d}{d x}\left(x^{2}\right)-(a+b) \frac{d}{d x}(x)+\frac{d}{d x}(a b)$ On using theorem $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$, we obtain $f^{\prime}(x)=2 x-(a+b)+0=2 x-a-b$ (ii) Let $f(x)=\left(a x^{2}+b\right)^...
Read More →The total revenue in Rupees received from the sale of x units of a product is given by
Question: The total revenue in Rupees received from the sale ofxunits of a product is given by $R(x)=13 x^{2}+26 x+15$ Find the marginal revenue whenx= 7. Solution: Marginal revenue is the rate of change of total revenue with respect to the number of units sold. $\therefore$ Marginal Revenue $(\mathrm{MR})=\frac{d R}{d x}=13(2 x)+26=26 x+26$ Whenx= 7, $M R=26(7)+26=182+26=208$ Hence, the required marginal revenue is Rs 208....
Read More →The total cost C (x) in Rupees associated with the production of x units of an item is given by
Question: The total costC(x) in Rupees associated with the production ofxunits of an item is given by $C(x)=0.007 x^{3}-0.003 x^{2}+15 x+4000$ Find the marginal cost when 17 units are produced. Solution: Marginal cost is the rate of change of total cost with respect to output. $\therefore$ Marginal cost $(M C)=\frac{d C}{d x}=0.007\left(3 x^{2}\right)-0.003(2 x)+15$ $=0.021 x^{2}-0.006 x+15$ When $x=17, \mathrm{MC}=0.021\left(17^{2}\right)-0.006(17)+15$ $=0.021(289)-0.006(17)+15$ $=6.069-0.102+1...
Read More →Sand is pouring from a pipe at the rate of
Question: Sand is pouring from a pipe at the rate of $12 \mathrm{~cm}^{3} / \mathrm{s}$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4 \mathrm{~cm}$ ? Solution: The volume ofa cone (V)with radius (r) and height (h)is given by, $V=\frac{1}{3} \pi r^{2} h$ It is given that, $h=\frac{1}{6} r \Rightarrow r=6 h$ $\therefore V=\frac{1}{3} \pi(6...
Read More →Find the derivative of
Question: Find the derivative of $x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}$ for some fixed real number $a$. Solution: Let $f(x)=x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}$ $\therefore f^{\prime}(x)=\frac{d}{d x}\left(x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}\right)$ $=\frac{d}{d x}\left(x^{n}\right)+a \frac{d}{d x}\left(x^{n-1}\right)+a^{2} \frac{d}{d x}\left(x^{n-2}\right)+\ldots+a^{n-1} \frac{d}{d x}(x)+a^{n} \frac{d}{d x}(1)$ On using theorem $\frac{d}{d x} x^...
Read More →Are the following statements true or false? Give reasons for your answer.
Question: Are the following statements true or false? Give reasons for your answer.(i) Every whole number is a natural number.(ii) Every integer is a rational number.(iii) Every rational number is an integer.(iv) Every natural number is a whole number.(v) Every integer is a whole number.(vi) Every rational number is a whole number. Solution: (i) False, because whole numbers start from zero and natural numbers start from one (ii) True, because it can be written in the form of a fraction with deno...
Read More →A balloon, which always remains spherical,
Question: A balloon, which always remains spherical, has a variable diameter $\frac{3}{2}(2 x+1)$. Find the rate of change of its volume with respect to $x$. Solution: The volume ofa sphere (V) with radius (r) is given by, $V=\frac{4}{3} \pi r^{3}$ It is given that: Diameter $=\frac{3}{2}(2 x+1)$ $\Rightarrow r=\frac{3}{4}(2 x+1)$ $\therefore V=\frac{4}{3} \pi\left(\frac{3}{4}\right)^{3}(2 x+1)^{3}=\frac{9}{16} \pi(2 x+1)^{3}$ Hence, the rate of change of volume with respect toxis as $\frac{d V}...
Read More →For the function
Question: For the function $f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1$ Prove that $f^{\prime}(1)=100 f^{\prime}(0)$ Solution: The given function is $f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1$ $\frac{d}{d x} f(x)=\frac{d}{d x}\left[\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1\right]$ $\frac{d}{d x} f(x)=\frac{d}{d x}\left(\frac{x^{100}}{100}\right)+\frac{d}{d x}\left(\frac{x^{99}}{99}\right)+\ldots+\frac{d}{d x}\left(\fra...
Read More →Find five rational numbers between
Question: Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$. Solution: We need to find 5 rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$. Since, LCM of denominators $=L C M(5,5)=5$ So, consider $\frac{3}{5}=\frac{3}{5} \times \frac{6}{6}$ $\Rightarrow \frac{3}{5}=\frac{18}{30}$ And, $\Rightarrow \frac{4}{5}=\frac{4}{5} \times \frac{6}{6}$ $\Rightarrow \frac{4}{5}=\frac{24}{30}$...
Read More →Find the derivative of the following functions from first principle.
Question: Find the derivative of the following functions from first principle. (i) $x^{3}-27$ (ii) $(x-1)(x-2)$ (iii) $\frac{1}{x^{2}}$ (iv) $\frac{x+1}{x-1}$ Solution: (i) Let $f(x)=x^{3}-27$. Accordingly, from the first principle, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $=\lim _{h \rightarrow 0} \frac{\left[(x+h)^{3}-27\right]-\left(x^{3}-27\right)}{h}$ $=\lim _{h \rightarrow 0} \frac{x^{3}+h^{3}+3 x^{2} h+3 x h^{2}-x^{3}}{h}$ $=\lim _{h \rightarrow 0} \frac{h^{3}+3 x^{2}...
Read More →Find six rational numbers between 3 and 4.
Question: Find six rational numbers between 3 and 4. Solution: We need to find 6 rational numbers between 3and 4. Consider, $3=\frac{3}{1}$ $\Rightarrow 3=\frac{3}{1} \times \frac{7}{7}$ $\Rightarrow 3=\frac{21}{7}$ And $4=\frac{4}{1}$ $\Rightarrow 4=\frac{4}{1} \times \frac{7}{7}$ $\Rightarrow 4=\frac{28}{7}$...
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