Explanation Given: The provided function is: g ( x ) = x And, the derivative of the provided function is: g ′ ( x ) = 1 2 x . Formula used: The tangent line to the graph of y = f ( x ) at the point ( a , f ( a ) ) is the line through this point having slope: m = f ′ ( a ) The point-slope form of a line is given by: y − y 1 = m ( x − x 1 ) Where, m is the slope of the line and ( x 1 , y 1 ) is the point through which it passes. Calculation: Consider the provided function: g ( x ) = x …… (1) Now, the derivative of this function is provided to be: g ′ ( x ) = 1 2 x To calculate the slope of tangent line at point x = 9 , substitute 9 for x in the above equation to get: g ′ ( 9 ) = 1 2 9 = 1 6 Thus, the slope of required tangent line is m = 1 6 . Now, substitute 9 for x in equation (1). g ( 9 ) = 9 = 3 Thus, the required tangent line is passing through the point ( 9 , 3 )

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Chapter 11.4, Problem 13E

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To calculate: The equation of tangent line at x=9 to the graph of function g(x)=x.

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Chapter 11.4, Problem 12E

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Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.

************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.

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The equation of tangent line at x = 9 to the graph of function g ( x ) = x .

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