Calculus Concepts Quiz

Derivatives

• For the function ( f(x) = 3x^3 ), the derivative ( f'(x) ) is calculated using the power rule.
• The power rule states that ( \frac{d}{dx}(x^n) = n \cdot x^{n-1} ).
• Applying this to ( f(x) ):
  • ( f'(x) = 3 \cdot 3x^{3-1} = 9x^2 ).

Area Under a Curve

• The area under the curve of the function ( f(x) = 2x ) from ( x = 0 ) to ( x = 3 ) can be found using the definite integral.
• The appropriate integral representation is ( \int_{0}^{3} 2x , dx ).

Limits

• The limit of the function ( g(x) = \frac{1}{x} ) as ( x ) approaches infinity is determined as follows:
• As ( x ) increases, ( \frac{1}{x} ) approaches 0.
• Therefore, ( \lim_{x \to \infty} g(x) = 0 ).

Concavity of a Function

• To determine the concavity of a function, the second derivative test is utilized.
• If the second derivative ( f''(x) ) is positive, the function is concave up; if negative, it is concave down.

Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus links differentiation with integration.
• It consists of two parts:
  • The first part states that if ( f ) is continuous on ([a, b]), then ( F(x) = \int_{a}^{x} f(t) , dt ) is differentiable, and ( F'(x) = f(x) ).
  • The second part states that if ( f ) is continuous over ([a, b]), then the definite integral ( \int_{a}^{b} f(x) , dx ) can be computed using any antiderivative of ( f ).