બહુમુખી કલન (Multivariable Calculus)

પરિચય: બહુમુખી કલન એ ગણિતની એક શાખા છે, જે બે અથવા વધુ સ્વતંત્ર ચડાવણીઓ સાથે સંકળાયેલી ફંક્શન્સનું વિશ્લેષણ કરે છે. આમાં ગ્રાફિંગ, ડેરિવેટિવ્સ, ઇન્ટિગ્રેશન, અને ફંક્શનના વિધાઓનો સમાવેશ થાય છે.

મહત્વ: બહુમુખી કલનનો ઉપયોગ વૈજ્ઞાનિક, ઇજનેરી, અને અર્થશાસ્ત્રમાં કરવામાં આવે છે. આમાં અપાતી ફંક્શન્સને તપાસવા, અપેક્ષાઓને મૂલ્યવાન બનાવવા, અને વિવિધ એપ્લિકેશન્સમાં સમસ્યાઓને ઉકેલવા માટે ઉપયોગ થાય છે.

બહુમુખી ફંક્શન:

બહુમુખી ફંક્શન એ ફંક્શન છે જે અનેક ચડાવણીઓ લે છે. ઉદાહરણ તરીકે, $f(x,y)$ જ્યાં $x$ અને $y$ બે સ્વતંત્ર ચડાવણીઓ છે.

આધારભૂત સુત્રો:

1. ડેરિવેટિવ્સ:
   • બહુમુખી ફંક્શનનું ડેરિવેટિવ ગણવત્તાને દર્શાવે છે.
   • ઉદાહરણ: $\frac{\partial f}{\partial x}$ અને $\frac{\partial f}{\partial y}$
2. ગ્રેડિએન્ટ:
   • ગ્રેડિએન્ટ એ બિનરેખીય પ્રવાહના મિકેનિકલ ગુણધર્મોને દર્શાવે છે.
   • ઉદાહરણ: $\mathrm{\nabla }f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$
3. ડબલ ઇન્ટિગ્રલ:
   • એ અલગ-અલગ કાર્ય માટે વિસ્તારમાં મૂલ્યવાન હોવા માટેની ગણતરી છે.
   • ઉદાહરણ: ${\iint }_{D}f(x,y),dA$

ઉદાહરણો:

હવે 100 ઉદાહરણો આપવામાં આવ્યા છે, જેમાં દરેક ઉદાહરણ બહુમુખી કલનના વિવિધ તત્વોને દર્શાવે છે.

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Example 1:

પંક્તિ:

$f(x,y)=x^2+y^2$

Step 1: Given Data:

$x=3$
$y=4$

Step 2: Solution:

$f(3,4)=3^2+4^2$
$=9+16$
$=25$

Step 3: Final Answer:

$25$

---

Example 2:

પંક્તિ:

$g(x,y)=2xy+3y^2$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$g(2,3)=2(2)(3)+3(3^2)$
$=12+27$
$=39$

Step 3: Final Answer:

$39$

---

Example 3:

પંક્તિ:

$h(x,y)=x^3–3xy+y^3$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$h(1,2)=1^3–3(1)(2)+2^3$
$=1–6+8$
$=3$

Step 3: Final Answer:

$3$

---

Example 4:

પંક્તિ:

$p(x,y)=e^{xy}$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$p(1,1)=e^{1\cdot 1}$
$=e^1$
$=e$

Step 3: Final Answer:

$e$

---

Example 5:

પંક્તિ:

$q(x,y)=\sin (xy)+\cos (x+y)$

Step 1: Given Data:

$x=\frac{\pi }{2}$
$y=\frac{\pi }{2}$

Step 2: Solution:

$q(\frac{\pi }{2},\frac{\pi }{2})=\sin (\frac{\pi }{2}\cdot \frac{\pi }{2})+\cos (\frac{\pi }{2}+\frac{\pi }{2})$
$=\sin \left(\frac{{\pi }^2}{4}\right)+\cos (\pi )$
$=\sin \left(\frac{{\pi }^2}{4}\right)–1$

Step 3: Final Answer:

$\sin \left(\frac{{\pi }^2}{4}\right)–1$

---

Example 6:

પંક્તિ:

$r(x,y)=x^{2}y+x{y}^2$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$r(2,3)=(2^2)(3)+(2)(3^2)$
$=12+18$
$=30$

Step 3: Final Answer:

$30$

---

Example 7:

પંક્તિ:

$s(x,y)=\ln (xy)$

Step 1: Given Data:

$x=4$
$y=2$

Step 2: Solution:

$s(4,2)=\ln (4\cdot 2)$
$=\ln (8)$

Step 3: Final Answer:

$\ln (8)$

---

Example 8:

પંક્તિ:

$t(x,y)=x^4+y^4$

Step 1: Given Data:

$x=2$
$y=2$

Step 2: Solution:

$t(2,2)=2^4+2^4$
$=16+16$
$=32$

Step 3: Final Answer:

$32$

---

Example 9:

પંક્તિ:

$u(x,y)=x^{2}y–4y^2$

Step 1: Given Data:

$x=3$
$y=1$

Step 2: Solution:

$u(3,1)=3^2(1)–4(1^2)$
$=9–4$
$=5$

Step 3: Final Answer:

$5$

---

Example 10:

પંક્તિ:

$v(x,y)=\frac{x^2+y^2}{x+y}$

Step 1: Given Data:

$x=4$
$y=2$

Step 2: Solution:

$v(4,2)=\frac{4^2+2^2}{4+2}$
$=\frac{16+4}{6}$
$=\frac{20}{6}=\frac{10}{3}$

Step 3: Final Answer:

$\frac{10}{3}$

---

Example 11:

પંક્તિ:

$w(x,y)=x^3+y^3–3xy$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$w(1,2)=1^3+2^3–3(1)(2)$
$=1+8–6$
$=3$

Step 3: Final Answer:

$3$

---

Example 12:

પંક્તિ:

$z(x,y)=e^{x+y}$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$z(1,1)=e^{1+1}$
$=e^2$

Step 3: Final Answer:

$e^2$

---

Example 13:

પંક્તિ:

$a(x,y)=x^2–y^2$

Step 1: Given Data:

$x=5$
$y=3$

Step 2: Solution:

$a(5,3)=5^2–3^2$
$=25–9$
$=16$

Step 3: Final Answer:

$16$

---

Example 14:

પંક્તિ:

$b(x,y)=\sqrt{x^2+y^2}$

Step 1: Given Data:

$x=3$
$y=4$

Step 2: Solution:

$b(3,4)=\sqrt{3^2+4^2}$
$=\sqrt{9+16}$
$=\sqrt{25}=5$

Step 3: Final Answer:

$5$

---

Example 15:

પંક્તિ:

$c(x,y)=\tan (x+y)$

Step 1: Given Data:

$x=\frac{\pi }{4}$
$y=\frac{\pi }{4}$

Step 2: Solution:

$c(\frac{\pi }{4},\frac{\pi }{4})=\tan (\frac{\pi }{4}+\frac{\pi }{4})$
$=\tan \left(\frac{\pi }{2}\right)=\text{undefined}$

Step 3: Final Answer:

$\text{undefined}$

---

Example 16:

પંક્તિ:

$d(x,y)=x^2+2xy+y^2$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$d(1,2)=1^2+2(1)(2)+2^2$
$=1+4+4$
$=9$

Step 3: Final Answer:

$9$

---

Example 17:

પંક્તિ:

$e(x,y)=\ln (xy)$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$e(2,3)=\ln (2\cdot 3)$
$=\ln (6)$

Step 3: Final Answer:

$\ln (6)$

---

Example 18:

પંક્તિ:

$f(x,y)=x^3+y^3–3xy$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$f(2,1)=2^3+1^3–3(2)(1)$
$=8+1–6$
$=3$

Step 3: Final Answer:

$3$

---

Example 19:

પંક્તિ:

$g(x,y)=x^2–4y$

Step 1: Given Data:

$x=3$
$y=1$

Step 2: Solution:

$g(3,1)=3^2–4(1)$
$=9–4$
$=5$

Step 3: Final Answer:

$5$

---

Example 20:

પંક્તિ:

$h(x,y)=2x+3y^2$

Step 1: Given Data:

$x=2$
$y=2$

Step 2: Solution:

$h(2,2)=2(2)+3(2^2)$
$=4+12$
$=16$

Step 3: Final Answer:

$16$

---

Example 21:

પંક્તિ:

$i(x,y)=x^4+y^4$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$i(2,3)=2^4+3^4$
$=16+81$
$=97$

Step 3: Final Answer:

$97$

---

Example 22:

પંક્તિ:

$j(x,y)=x^2+2xy+y^2$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$j(1,1)=1^2+2(1)(1)+1^2$
$=1+2+1$
$=4$

Step 3: Final Answer:

$4$

---

Example 23:

પંક્તિ:

$k(x,y)=\frac{x+y}{xy}$

Step 1: Given Data:

$x=2$
$y=5$

Step 2: Solution:

$k(2,5)=\frac{2+5}{2\cdot 5}$
$=\frac{7}{10}$

Step 3: Final Answer:

$\frac{7}{10}$

---

Example 24:

પંક્તિ:

$l(x,y)=x^2–2y+3$

Step 1: Given Data:

$x=4$
$y=1$

Step 2: Solution:

$l(4,1)=4^2–2(1)+3$
$=16–2+3$
$=17$

Step 3: Final Answer:

$17$

---

Example 25:

પંક્તિ:

$m(x,y)=3x^2+2y^2–xy$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$m(2,3)=3(2^2)+2(3^2)–(2)(3)$
$=3(4)+2(9)–6$
$=12+18–6$
$=24$

Step 3: Final Answer:

$24$

---

Example 26:

પંક્તિ:

$n(x,y)=\sqrt{x^2+y^2}$

Step 1: Given Data:

$x=3$
$y=4$

Step 2: Solution:

$n(3,4)=\sqrt{3^2+4^2}$
$=\sqrt{9+16}$
$=\sqrt{25}=5$

Step 3: Final Answer:

$5$

---

Example 27:

પંક્તિ:

$o(x,y)=\cos (x)+\sin (y)$

Step 1: Given Data:

$x=0$
$y=\frac{\pi }{2}$

Step 2: Solution:

$o(0,\frac{\pi }{2})=\cos (0)+\sin \left(\frac{\pi }{2}\right)$
$=1+1$
$=2$

Step 3: Final Answer:

$2$

---

Example 28:

પંક્તિ:

$p(x,y)=x^2–2xy+y^2$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$p(2,1)=2^2–2(2)(1)+1^2$
$=4–4+1$
$=1$

Step 3: Final Answer:

$1$

---

Example 29:

પંક્તિ:

$q(x,y)=\tan (x+y)$

Step 1: Given Data:

$x=\frac{\pi }{4}$
$y=\frac{\pi }{4}$

Step 2: Solution:

$q(\frac{\pi }{4},\frac{\pi }{4})=\tan (\frac{\pi }{4}+\frac{\pi }{4})$
$=\tan \left(\frac{\pi }{2}\right)=\text{undefined}$

Step 3: Final Answer:

$\text{undefined}$

---

Example 30:

પંક્તિ:

$r(x,y)=e^{x^2+y^2}$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$r(1,1)=e^{1^2+1^2}$
$=e^2$

Step 3: Final Answer:

$e^2$

---

Example 31:

પંક્તિ:

$s(x,y)=\ln (x+y)$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$s(2,3)=\ln (2+3)$
$=\ln (5)$

Step 3: Final Answer:

$\ln (5)$

---

Example 32:

પંક્તિ:

$t(x,y)=x^3+3xy+y^3$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$t(1,1)=1^3+3(1)(1)+1^3$
$=1+3+1$
$=5$

Step 3: Final Answer:

$5$

---

Example 33:

પંક્તિ:

$u(x,y)=x^2+4xy+4y^2$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$u(2,1)=2^2+4(2)(1)+4(1^2)$
$=4+8+4$
$=16$

Step 3: Final Answer:

$16$

---

Example 34:

પંક્તિ:

$v(x,y)=x^{2}y–y^{2}x$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$v(2,3)=2^2(3)–3^2(2)$
$=12–18$
$=-6$

Step 3: Final Answer:

$-6$

---

Example 35:

પંક્તિ:

$w(x,y)=3x–2y+xy$

Step 1: Given Data:

$x=5$
$y=2$

Step 2: Solution:

$w(5,2)=3(5)–2(2)+(5)(2)$
$=15–4+10$
$=21$

Step 3: Final Answer:

$21$

---

Example 36:

પંક્તિ:

$x(x,y)=x^2+y^2+2xy$

Step 1: Given Data:

$x=1$
$y=3$

Step 2: Solution:

$x(1,3)=1^2+3^2+2(1)(3)$
$=1+9+6$
$=16$

Step 3: Final Answer:

$16$

---

Example 37:

પંક્તિ:

$y(x,y)=\frac{1}{x}+\frac{1}{y}$

Step 1: Given Data:

$x=2$
$y=4$

Step 2: Solution:

$y(2,4)=\frac{1}{2}+\frac{1}{4}$
$=\frac{2}{4}+\frac{1}{4}$
$=\frac{3}{4}$

Step 3: Final Answer:

$\frac{3}{4}$

---

Example 38:

પંક્તિ:

$z(x,y)=x^3+y^3–3xy$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$z(1,2)=1^3+2^3–3(1)(2)$
$=1+8–6$
$=3$

Step 3: Final Answer:

$3$

---

Example 39:

પંક્તિ:

$a(x,y)=e^{x+y}–x^2{y}^2$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$a(1,1)=e^{1+1}–(1^2)(1^2)$
$=e^2–1$

Step 3: Final Answer:

$e^2–1$

---

Example 40:

પંક્તિ:

$b(x,y)=\cos (x)+\sin (y)$

Step 1: Given Data:

$x=0$
$y=\frac{\pi }{2}$

Step 2: Solution:

$b(0,\frac{\pi }{2})=\cos (0)+\sin \left(\frac{\pi }{2}\right)$
$=1+1$
$=2$

Step 3: Final Answer:

$2$

---

Example 41:

પંક્તિ:

$c(x,y)=\frac{x^2+y^2}{xy}$

Step 1: Given Data:

$x=2$
$y=4$

Step 2: Solution:

$c(2,4)=\frac{2^2+4^2}{2\cdot 4}$
$=\frac{4+16}{8}$
$=\frac{20}{8}=\frac{5}{2}$

Step 3: Final Answer:

$\frac{5}{2}$

---

Example 42:

પંક્તિ:

$d(x,y)=x^4+y^4–4xy(x^2+y^2)$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$d(1,1)=1^4+1^4–4(1)(1)(1^2+1^2)$
$=1+1–4(1)(2)$
$=2–8$
$=-6$

Step 3: Final Answer:

$-6$

---

Example 43:

પંક્તિ:

$e(x,y)=\sqrt{x^2+4y^2}$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$e(2,3)=\sqrt{2^2+4(3^2)}$
$=\sqrt{4+36}$
$=\sqrt{40}=2\sqrt{10}$

Step 3: Final Answer:

$2\sqrt{10}$

---

Example 44:

પંક્તિ:

$f(x,y)=x^2+3xy+y^2$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$f(1,2)=1^2+3(1)(2)+2^2$
$=1+6+4$
$=11$

Step 3: Final Answer:

$11$

---

Example 45:

પંક્તિ:

$g(x,y)=\tan (xy)$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$g(1,1)=\tan (1\cdot 1)$
$=\tan (1)$

Step 3: Final Answer:

$\tan (1)$

---

Example 46:

પંક્તિ:

$h(x,y)=\frac{x^2–y^2}{x+y}$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$h(2,1)=\frac{2^2–1^2}{2+1}$
$=\frac{4–1}{3}$
$=\frac{3}{3}=1$

Step 3: Final Answer:

$1$

---

Example 47:

પંક્તિ:

$i(x,y)=e^x\cdot \cos (y)$

Step 1: Given Data:

$x=0$
$y=0$

Step 2: Solution:

$i(0,0)=e^0\cdot \cos (0)$
$=1\cdot 1$
$=1$

Step 3: Final Answer:

$1$

---

Example 48:

પંક્તિ:

$j(x,y)=x+y+xy$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$j(2,3)=2+3+(2)(3)$
$=2+3+6$
$=11$

Step 3: Final Answer:

$11$

---

Example 49:

પંક્તિ:

$k(x,y)=x^{2}y–2x{y}^2$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$k(2,1)=2^2(1)–2(2)(1^2)$
$=4–4$
$=0$

Step 3: Final Answer:

$0$

---

Example 50:

પંક્તિ:

$l(x,y)=\frac{x^2+y^2}{2}$

Step 1: Given Data:

$x=4$
$y=2$

Step 2: Solution:

$l(4,2)=\frac{4^2+2^2}{2}$
$=\frac{16+4}{2}$
$=\frac{20}{2}=10$

Step 3: Final Answer:

$10$

---

Example 51:

પંક્તિ:

$m(x,y)=e^{xy}+x^2–y^2$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$m(1,1)=e^{1\cdot 1}+1^2–1^2$
$=e+1–1$
$=e$

Step 3: Final Answer:

$e$

---

Example 52:

પંક્તિ:

$n(x,y)=\frac{\sin (x)}{\cos (y)}$

Step 1: Given Data:

$x=\frac{\pi }{2}$
$y=0$

Step 2: Solution:

$n(\frac{\pi }{2},0)=\frac{\sin \left(\frac{\pi }{2}\right)}{\cos (0)}$
$=\frac{1}{1}$
$=1$

Step 3: Final Answer:

$1$

---

Example 53:

પંક્તિ:

$o(x,y)=x^2+2xy+y^2–4$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$o(1,1)=1^2+2(1)(1)+1^2–4$
$=1+2+1–4$
$=0$

Step 3: Final Answer:

$0$

---

Example 54:

પંક્તિ:

$p(x,y)=e^{x+y}–x^2{y}^2$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$p(2,3)=e^{2+3}–(2^2)(3^2)$
$=e^5–4\cdot 9$
$=e^5–36$

Step 3: Final Answer:

$e^5–36$

---

Example 55:

પંક્તિ:

$q(x,y)=\ln (xy)$

Step 1: Given Data:

$x=3$
$y=2$

Step 2: Solution:

$q(3,2)=\ln (3\cdot 2)$
$=\ln (6)$

Step 3: Final Answer:

$\ln (6)$

---

Example 56:

પંક્તિ:

$r(x,y)=x^3+y^3–3xy$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$r(2,1)=2^3+1^3–3(2)(1)$
$=8+1–6$
$=3$

Step 3: Final Answer:

$3$

---

Example 57:

પંક્તિ:

$s(x,y)=e^x+e^y$

Step 1: Given Data:

$x=1$
$y=0$

Step 2: Solution:

$s(1,0)=e^1+e^0$
$=e+1$

Step 3: Final Answer:

$e+1$

---

Example 58:

પંક્તિ:

$t(x,y)=x^{2}y+x{y}^2$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$t(2,3)=(2^2)(3)+(2)(3^2)$
$=12+18$
$=30$

Step 3: Final Answer:

$30$

---

Example 59:

પંક્તિ:

$u(x,y)=\sqrt{x^2+y^2}$

Step 1: Given Data:

$x=3$
$y=4$

Step 2: Solution:

$u(3,4)=\sqrt{3^2+4^2}$
$=\sqrt{9+16}$
$=\sqrt{25}=5$

Step 3: Final Answer:

$5$

---

Example 60:

પંક્તિ:

$v(x,y)=x^2+y^2+2xy$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$v(2,1)=2^2+1^2+2(2)(1)$
$=4+1+4$
$=9$

Step 3: Final Answer:

$9$

---

Example 61:

પંક્તિ:

$w(x,y)=x+y+xy$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$w(1,2)=1+2+(1)(2)$
$=1+2+2$
$=5$

Step 3: Final Answer:

$5$

---

Example 62:

પંક્તિ:

$x(x,y)=e^{x+y}–x^2{y}^2$

Step 1: Given Data:

$x=0$
$y=1$

Step 2: Solution:

$x(0,1)=e^{0+1}–(0^2)(1^2)$
$=e–0$
$=e$

Step 3: Final Answer:

$e$

---

Example 63:

પંક્તિ:

$y(x,y)=\frac{1}{x^2+y^2}$

Step 1: Given Data:

$x=2$
$y=2$

Step 2: Solution:

$y(2,2)=\frac{1}{2^2+2^2}$
$=\frac{1}{4+4}$
$=\frac{1}{8}$

Step 3: Final Answer:

$\frac{1}{8}$

---

Example 64:

પંક્તિ:

$z(x,y)=x^3–3x{y}^2$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$z(1,1)=1^3–3(1)(1^2)$
$=1–3$
$=-2$

Step 3: Final Answer:

$-2$

---

Example 65:

પંક્તિ:

$a(x,y)=e^x+\ln (y)$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$a(1,1)=e^1+\ln (1)$
$=e+0$
$=e$

Step 3: Final Answer:

$e$

---

Example 66:

પંક્તિ:

$b(x,y)=x^2+y^2–xy$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$b(2,1)=2^2+1^2–(2)(1)$
$=4+1–2$
$=3$

Step 3: Final Answer:

$3$

---

Example 67:

પંક્તિ:

$c(x,y)=x^{2}y+3x{y}^2$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$c(2,3)=(2^2)(3)+3(2)(3^2)$
$=12+54$
$=66$

Step 3: Final Answer:

$66$

---

Example 68:

પંક્તિ:

$d(x,y)=\frac{x^3+y^3}{xy}$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$d(2,3)=\frac{2^3+3^3}{2\cdot 3}$
$=\frac{8+27}{6}$
$=\frac{35}{6}$

Step 3: Final Answer:

$\frac{35}{6}$

---

Example 69:

પંક્તિ:

$e(x,y)=x^2+y^2+2xy–1$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$e(2,3)=2^2+3^2+2(2)(3)–1$
$=4+9+12–1$
$=24$

Step 3: Final Answer:

$24$

---

Example 70:

પંક્તિ:

$f(x,y)=\sqrt{x^2+y^2}$

Step 1: Given Data:

$x=5$
$y=12$

Step 2: Solution:

$f(5,12)=\sqrt{5^2+{12}^2}$
$=\sqrt{25+144}$
$=\sqrt{169}=13$

Step 3: Final Answer:

$13$

---

Example 71:

પંક્તિ:

$g(x,y)=3x+4y–xy$

Step 1: Given Data:

$x=3$
$y=2$

Step 2: Solution:

$g(3,2)=3(3)+4(2)–(3)(2)$
$=9+8–6$
$=11$

Step 3: Final Answer:

$11$

---

Example 72:

પંક્તિ:

$h(x,y)=x^2–4x+y^2–4y+4$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$h(2,3)=2^2–4(2)+3^2–4(3)+4$
$=4–8+9–12+4$
$=-3$

Step 3: Final Answer:

$-3$

---

Example 73:

પંક્તિ:

$i(x,y)=x^2{y}^2–2xy+1$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$i(1,1)=(1^2)(1^2)–2(1)(1)+1$
$=1–2+1$
$=0$

Step 3: Final Answer:

$0$

---

Example 74:

પંક્તિ:

$j(x,y)=e^{x+y}–2xy$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$j(1,2)=e^{1+2}–2(1)(2)$
$=e^3–4$

Step 3: Final Answer:

$e^3–4$

---

Example 75:

પંક્તિ:

$k(x,y)=x^3–3x{y}^2+2y^3$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$k(1,1)=1^3–3(1)(1^2)+2(1^3)$
$=1–3+2$
$=0$

Step 3: Final Answer:

$0$

---

Example 76:

પંક્તિ:

$l(x,y)=\frac{x^2+y^2}{2}$

Step 1: Given Data:

$x=2$
$y=2$

Step 2: Solution:

$l(2,2)=\frac{2^2+2^2}{2}$
$=\frac{4+4}{2}$
$=\frac{8}{2}=4$

Step 3: Final Answer:

$4$

---

Example 77:

પંક્તિ:

$m(x,y)=3x^2–2y^2+xy$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$m(1,2)=3(1^2)–2(2^2)+(1)(2)$
$=3–8+2$
$=-3$

Step 3: Final Answer:

$-3$

---

Example 78:

પંક્તિ:

$n(x,y)=x^2+y^2–2xy$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$n(2,3)=2^2+3^2–2(2)(3)$
$=4+9–12$
$=1$

Step 3: Final Answer:

$1$

---

Example 79:

પંક્તિ:

$o(x,y)=e^x\cdot \ln (y)$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$o(1,2)=e^1\cdot \ln (2)$
$=e\cdot \ln (2)$

Step 3: Final Answer:

$e\cdot \ln (2)$

---

Example 80:

પંક્તિ:

$p(x,y)=5x–3y+xy$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$p(2,3)=5(2)–3(3)+(2)(3)$
$=10–9+6$
$=7$

Step 3: Final Answer:

$7$

---

Example 81:

પંક્તિ:

$q(x,y)=x^{2}y+x{y}^2–2y^3$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$q(1,2)=(1^2)(2)+(1)(2^2)–2(2^3)$
$=1\cdot 2+1\cdot 4–2\cdot 8$
$=2+4–16$
$=-10$

Step 3: Final Answer:

$-10$

---

Example 82:

પંક્તિ:

$r(x,y)=\sqrt{x^2+3y^2}$

Step 1: Given Data:

$x=3$
$y=1$

Step 2: Solution:

$r(3,1)=\sqrt{3^2+3(1^2)}$
$=\sqrt{9+3}$
$=\sqrt{12}=2\sqrt{3}$

Step 3: Final Answer:

$2\sqrt{3}$

---

Example 83:

પંક્તિ:

$s(x,y)=\tan (x)+\tan (y)$

Step 1: Given Data:

$x=\frac{\pi }{4}$
$y=\frac{\pi }{4}$

Step 2: Solution:

$s(\frac{\pi }{4},\frac{\pi }{4})=\tan \left(\frac{\pi }{4}\right)+\tan \left(\frac{\pi }{4}\right)$
$=1+1$
$=2$

Step 3: Final Answer:

$2$

---

Example 84:

પંક્તિ:

$t(x,y)=x^2+y^2–xy+1$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$t(2,3)=2^2+3^2–(2)(3)+1$
$=4+9–6+1$
$=8$

Step 3: Final Answer:

$8$

---

Example 85:

પંક્તિ:

$u(x,y)=x^{2}y+4y^2–2x^3$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$u(1,1)=(1^2)(1)+4(1^2)–2(1^3)$
$=1+4–2$
$=3$

Step 3: Final Answer:

$3$

---

Example 86:

પંક્તિ:

$v(x,y)=e^{x+y}+x^2–y^2$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$v(1,2)=e^{1+2}+1^2–2^2$
$=e^3+1–4$
$=e^3–3$

Step 3: Final Answer:

$e^3–3$

---

Example 87:

પંક્તિ:

$w(x,y)=x^3–3x+y^3–3y$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$w(1,1)=1^3–3(1)+1^3–3(1)$
$=1–3+1–3$
$=-4$

Step 3: Final Answer:

$-4$

---

Example 88:

પંક્તિ:

$x(x,y)=\frac{xy+1}{x+y}$

Step 1: Given Data:

$x=2$
$y=3$

Step 2: Solution:

$x(2,3)=\frac{2(3)+1}{2+3}$
$=\frac{6+1}{5}$
$=\frac{7}{5}$

Step 3: Final Answer:

$\frac{7}{5}$

---

Example 89:

પંક્તિ:

$y(x,y)=\ln (xy)+x^2+y^2$

Step 1: Given Data:

$x=2$
$y=2$

Step 2: Solution:

$y(2,2)=\ln (2\cdot 2)+2^2+2^2$
$=\ln (4)+4+4$
$=\ln (4)+8$

Step 3: Final Answer:

$\ln (4)+8$

---

Example 90:

પંક્તિ:

$z(x,y)=x^2+2xy+y^2$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$z(1,1)=1^2+2(1)(1)+1^2$
$=1+2+1$
$=4$

Step 3: Final Answer:

$4$

---

Example 91:

પંક્તિ:

$a(x,y)=x^3+y^3–3xy$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$a(2,1)=2^3+1^3–3(2)(1)$
$=8+1–6$
$=3$

Step 3: Final Answer:

$3$

---

Example 92:

પંક્તિ:

$b(x,y)=e^x+y^2–xy$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$b(1,2)=e^1+2^2–1(2)$
$=e+4–2$
$=e+2$

Step 3: Final Answer:

$e+2$

---

Example 93:

પંક્તિ:

$c(x,y)=x^4+y^4–4x^2{y}^2$

Step 1: Given Data:

$x=2$
$y=2$

Step 2: Solution:

$c(2,2)=2^4+2^4–4(2^2)(2^2)$
$=16+16–16$
$=16$

Step 3: Final Answer:

$16$

---

Example 94:

પંક્તિ:

$d(x,y)=x+y+\frac{1}{xy}$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$d(1,1)=1+1+\frac{1}{1\cdot 1}$
$=1+1+1$
$=3$

Step 3: Final Answer:

$3$

---

Example 95:

પંક્તિ:

$e(x,y)=x^3+2xy+y^3–6$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$e(2,1)=2^3+2(2)(1)+1^3–6$
$=8+4+1–6$
$=7$

Step 3: Final Answer:

$7$

---

Example 96:

પંક્તિ:

$f(x,y)=\frac{x^2–y^2}{x+y}$

Step 1: Given Data:

$x=3$
$y=1$

Step 2: Solution:

$f(3,1)=\frac{3^2–1^2}{3+1}$
$=\frac{9–1}{4}$
$=\frac{8}{4}=2$

Step 3: Final Answer:

$2$

---

Example 97:

પંક્તિ:

$g(x,y)=e^{xy}+x^{2}y–2y^2$

Step 1: Given Data:

$x=1$
$y=2$

Step 2: Solution:

$g(1,2)=e^{1\cdot 2}+(1^2)(2)–2(2^2)$
$=e^2+2–8$
$=e^2–6$

Step 3: Final Answer:

$e^2–6$

---

Example 98:

પંક્તિ:

$h(x,y)=2x+3y–xy$

Step 1: Given Data:

$x=2$
$y=1$

Step 2: Solution:

$h(2,1)=2(2)+3(1)–(2)(1)$
$=4+3–2$
$=5$

Step 3: Final Answer:

$5$

---

Example 99:

પંક્તિ:

$i(x,y)=x^2+y^2–4xy+1$

Step 1: Given Data:

$x=1$
$y=1$

Step 2: Solution:

$i(1,1)=1^2+1^2–4(1)(1)+1$
$=1+1–4+1$
$=-1$

Step 3: Final Answer:

$-1$

---

Example 100:

પંક્તિ:

$j(x,y)=e^{x+y}+xy–1$

Step 1: Given Data:

$x=0$
$y=0$

Step 2: Solution:

$j(0,0)=e^{0+0}+(0)(0)–1$
$=e^0+0–1$
$=1+0–1$
$=0$

Step 3: Final Answer:

$0$