Explanation

$$lim_{x to 0} frac{1 - cos x}{x^2} = lim_{x to 0} frac{sin x}{2x} = frac{1}{2}$$

Explanation

Three axioms define a topology:

The empty set and the whole space must be open. 

The union of any collection of open sets must be open. 

The intersection of a finite number of open sets must be open. 

Why is option B incorrect?

While the intersection of a finite number of open sets is open, the intersection of an infinite number of open sets is not necessarily open.

This is a key distinction in the definition of a topology. 

Explanation

Explanation

What is the value of the integral:

∫e^x2+lnx dx

Step 1: Simplify the integrand

Recall:

e^a+b=e^a⋅e^b

So,

e^x^2+lnx=e^x^2⋅e^lnx=x⋅e^x^2

Step 2: The integral becomes:

∫x⋅e^x^2 dx

Now use substitution:

Let $u = x^2 Rightarrow du = 2x , dx Rightarrow frac{1}{2} du = x , dx$

So,

$$int x cdot e^{x^2} , dx = frac{1}{2} int e^u , du = frac{1}{2} e^u + C = frac{1}{2} e^{x^2} + C$$

=1​/2∫e^u du

=1​/2e^u+C

=1​/2ex^2 +C

Explanation

We use the product rule since it's the product of two functions:

$$f(x) = x^3 cdot ln(x)$$

Let:

• $u = x^3 Rightarrow u' = 3x^2$

• $v = ln(x) Rightarrow v' = frac{1}{x}$

Apply the product rule:

f′(x)=u′v+uv′=3x^2⋅ln(x)+x^3⋅1/x​
f′(x)=3^x2ln(x)+x^2

Explanation

The expression p ∨ ¬p (read as "p or not p") is a tautology.

A tautology is a logical statement that is always true regardless of the truth value of its components.

 p ∨ ¬p is always true because either p is true or ¬p (not p) is true.

Explanation

 In the equation Y = a + bX, "b" represents the Slope of the line. 

The slope-intercept form of a linear equation is y = mx + b, where "m" is the slope and "b" is the y-intercept. 

Explanation

A one-one function means that each element of the domain maps to a unique component of the codomain.

This is the definition of an injective function.

A bijective function is both injective and surjective.

A surjective function covers every element in the codomain.

A constant function maps all domain elements to the same codomain value — it is not injective.

Explanation

About vpasolve():

It often requires an initial guess to start the numerical search. MathWorks notes this 

It can be used to solve systems of equations. 

It may return only one solution, depending on the equation and parameters

Explanation

In topology, isometric metric spaces represent an equivalence relation. 

An equivalence relation needs to be reflexive, symmetric, and transitive.