. Question 8 Hint: Enter the tangent line as an equation using x and y. Let f(a) = 4ac2 + 13x - 3. Using the definition of derivative, f' (x) = lim f(a t h) - f(2) h -0 h enter the expression needed to find the derivative function. f' (x) = lim h -+0 df After evaluating this limit, we see that f' (ac) = dx Finally, the equation of the tangent line to f (a) where x = 3 issolution : Given that, fin = 4x"+ 13 2 - 3 Now , we have to find the derivative of for by useing the wefination of derivative. . The derivative f' (x ) is defined as : f ( x ) = lim flath ) - f(x ) ho h - f ( x ) - aim (4 ( ath ) + 13 (xth )- 3) - (4x" + 13x-3) b So, f ( x ) = 1 aim (4 ( *+6 ) + 13 ( 2+ 6 ) - 3 ) - (4x4 + 13x - 3 ) ho h . Now evaluating the limit f'() = Jim (4 ( xth ) + 13 ( 2th) - 3) - (42 +1321-3) h 4 ( 2 + 2 x h + hy ) + 13 * + 134 - 3 - 4 21 8 - 13 * + 3 h lim 4 x + 8 xch + 4 h + 1321 + 13h - 3 - 4x - 13 21 + 3 h 8 xh +4h +13h hoo h = (8* + 4h + 13 ) = 8 x + 4 x0 + 13 82 + 13 so, f ' ( x ) - = 8x + 13. Now , we have to determine the equation of the tangent line to fix, at n= 3 . We know that , the equation of a tangent line to few at n = a is 4 - flaj = f'(a) ( x1 - a ) NOW, f (3) = 1 ( 3 ) + 13 (3 ) - 3 = 4x9 + 39 - 3 2 36 + 39 - 3 = 72 and f' ( 3 ) = 8 (3 ) + 13 = 24+ 13 = 37 There fore, the equation of the tangent line to fix at 2 = 3 is : - 72 = 37 (2-3) " f ( 3 ) = 72 f ' ( 3 ) = 37 4- 72 = 37x -111 4 = 372 - 111+ 72 => 8 = 372 - 39