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Compute the derivative of the function f(x)=x2+3x+5f(x) = x^2 + 3x + 5.
Dec 14, 2023
Compute the derivative of the function f(x)=x2+3x+5f(x) = x^2 + 3x + 5.
Solution by Steps
step 1
To compute the derivative of the function f(x)=x2+3x+5 f(x) = x^2 + 3x + 5 , we will use the power rule for differentiation, which states that the derivative of xn x^n is nxn−1 nx^{n-1}
step 2
Differentiating the term x2 x^2 using the power rule: ddxx2=2x \frac{d}{dx}x^2 = 2x
step 3
Differentiating the term 3x 3x using the power rule: ddx3x=3 \frac{d}{dx}3x = 3
step 4
The derivative of a constant is 0, so for the term 5 5 : ddx5=0 \frac{d}{dx}5 = 0
step 5
Combining the derivatives from steps 2, 3, and 4 to find the derivative of the entire function: f′(x)=2x+3+0 f'(x) = 2x + 3 + 0
Answer
f′(x)=2x+3 f'(x) = 2x + 3
Key Concept
Power Rule for Differentiation
Explanation
The derivative of f(x)=x2+3x+5 f(x) = x^2 + 3x + 5 is found by applying the power rule to each term separately and then combining the results.
(6) Determine whether each statement is true or false with justification. (i) A set of vectors {v1,v2,…,vk}\left\{v_{1}, v_{2}, \ldots, v_{k}\right\} in a vector space is called linearly dependent if the vector equation c1v1+c2v2+⋯+ckvk=0c_{1} v_{1}+c_{2} v_{2}+\cdots+c_{k} v_{k}=0 has only the trivial solution. (ii) Two vectors and in a vector space are linearly dependent if and only if one is a scalar multiple of the other. (iii) A set is {v1,v2,…,vk},k≥2\left\{v_{1}, v_{2}, \ldots, v_{k}\right\}, k \geq 2 linearly independent if and only if at least one of the vectors can be written as a linear combination of the other vectors.
Solution by Steps
step 1
To address the first statement, we recall the definition of linear dependence. A set of vectors is linearly dependent if there exists a non-trivial solution to the equation c1v1+c2v2+⋯+ckvk=0c_{1} v_{1} + c_{2} v_{2} + \cdots + c_{k} v_{k} = 0. A non-trivial solution means that not all cic_i are zero
step 2
The statement is false because it incorrectly defines linear dependence. If the equation has only the trivial solution (where all cic_i are zero), then the vectors are actually linearly independent, not dependent
Answer
The statement (i) is false.
Key Concept
Linear dependence and independence
Explanation
A set of vectors is linearly dependent if there exists a non-trivial solution to the vector equation where the coefficients are not all zero.
step 1
To address the second statement, we consider the definition of linear dependence between two vectors. Two vectors are linearly dependent if and only if one can be expressed as a scalar multiple of the other
step 2
The statement is true because if two vectors are linearly dependent, one vector can be written as a scalar multiple of the other, and vice versa
Answer
The statement (ii) is true.
Key Concept
Linear dependence of two vectors
Explanation
Two vectors are linearly dependent if and only if one is a scalar multiple of the other, which is the definition of linear dependence for two vectors.
step 1
To address the third statement, we recall the definition of linear independence. A set of vectors is linearly independent if the only solution to the equation c1v1+c2v2+⋯+ckvk=0c_{1} v_{1} + c_{2} v_{2} + \cdots + c_{k} v_{k} = 0 is the trivial solution, where all cic_i are zero
step 2
The statement is false because it incorrectly defines linear independence. If at least one of the vectors can be written as a linear combination of the others, then the set is actually linearly dependent
Answer
The statement (iii) is false.
Key Concept
Linear independence of a set of vectors
Explanation
A set of vectors is linearly independent if the only solution to the vector equation is the trivial solution, not if one of the vectors can be written as a linear combination of the others.
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