Quiz: LOGARITHMIC LIMITS AND DERIVATIVES — 8 domande

1. What is the derivative of the product of two differentiable functions called?

A. Product rule

B. Chain rule

C. Power rule

D. Quotient rule

Answer: (A)

Spiegazione

The derivative of the product of two functions is given by the product rule, which states that (uv)' = u'v + uv'. This rule specifically describes how to differentiate a product of functions, making it the correct answer.

2. What is the primitive (antiderivative) of the function $rac{u'}{u}$, where $u$ is differentiable and non-zero?

A. $u$

B. $rac{1}{u}$

C. $rac{u'}{u}$

D. $ ext{ln}|u|$

Answer: (D)

Spiegazione

The primitive of the function $rac{u'}{u}$, where $u$ is differentiable and non-zero, is $ ext{ln}|u| + C$. This is a standard result in calculus, derived from the chain rule for the logarithm function.

3. What is the role or purpose of understanding the limit of the logarithm function as its argument approaches zero from the right?

A. To recognize that ln x tends to positive infinity as x approaches zero from the right.

B. To understand that ln x tends to negative infinity as x approaches zero from the right.

C. To find the derivative of ln x at zero.

D. To determine the value of ln x at zero.

Answer: (B)

Spiegazione

The limit of ln x as x approaches zero from the right is -∞, which helps in analyzing the behavior of functions involving logarithms near zero, especially in limits and asymptotic analysis.

4. When was the limit of the logarithm at infinity, specifically that $ ext{ln } x o + ext{infinity}$ as $x o + ext{infinity}$, established or recognized?

A. During the development of calculus in the late 17th century

B. In the 15th century during the Renaissance

C. In the 19th century with the formalization of real analysis

D. In the early 20th century with modern analysis

Answer: (A)

Spiegazione

The limit of $ ext{ln } x$ as $x o + ext{infinity}$ was established during the development of calculus in the late 17th century, when mathematicians like Newton and Leibniz formalized the properties of logarithms and limits.

5. How are the concepts of the limit of $ ext{ln} x$ at zero and at infinity similar or different?

A. Both limits tend to infinity, but in opposite directions.

B. Both limits tend to positive infinity.

C. Both limits tend to finite values, but at different points.

D. Both limits tend to negative infinity.

Answer: (A)

Spiegazione

Both limits involve the logarithm function approaching unbounded values at boundary points, with $ ext{ln} x o -inite$ as $x o 0^+$ and $ ext{ln} x o +inite$ as $x o +inite$. They are similar in that both are unbounded, but differ in the direction of divergence.

6. Who is credited with formulating the fundamental properties of the natural logarithm, including its primitive and relation to derivatives such as u'/u?

A. Gottfried Wilhelm Leibniz

B. Carl Friedrich Gauss

C. Leonhard Euler

D. Isaac Newton

Answer: (C)

Spiegazione

Leonhard Euler is credited with formalizing the properties of the natural logarithm, including its primitive function and its relation to derivatives of ratios like u'/u. His work in the 18th century established these foundational concepts in calculus.

7. What is the effect on the limit of $ ext{ln}(U(x))$ when the argument $U(x)$ approaches zero from the right?

A. The limit tends to positive infinity

B. The limit tends to zero

C. The limit tends to a finite constant

D. The limit tends to negative infinity

Answer: (D)

Spiegazione

When $U(x)$ approaches zero from the right, $ ext{ln}(U(x))$ tends to negative infinity. This is because the natural logarithm function tends to $- ext{infinity}$ as its argument approaches zero from the positive side, reflecting the unbounded decrease of the logarithm near zero.

8. How can you evaluate the limit of the expression \\frac{ ext{ln}(x^2 + 1)}{x} as x approaches infinity?

A. The limit is 0 because \\text{ln}(x^2 + 1) grows slower than x.

B. The limit is 1 because the numerator and denominator grow at the same rate.

C. The limit does not exist because the numerator oscillates.

D. The limit is infinity because \\text{ln}(x^2 + 1) increases without bound.

Answer: (A)

Spiegazione

As x approaches infinity, \\text{ln}(x^2 + 1) behaves like \\text{ln}(x^2) = 2\\text{ln} x, which grows slower than x. Therefore, the ratio \\frac{ ext{ln}(x^2 + 1)}{x} tends to 0, applying the limit laws and properties of logarithms.