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Questions and Answers
What is the derivative of the function $f(x) = 3x^4 - 5x^3 + 2x - 7$?
Which of the following expressions represents the limit as $x$ approaches 2 for the function $ rac{x^2 - 4}{x - 2}$?
In a geometric sequence where the first term is 3 and the common ratio is 2, what is the 5th term?
If $x$ is a root of the polynomial $2x^3 - 7x^2 + 4x - 1$, what is the potential value of $x$ based on the Rational Root Theorem?
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What is the area of a circle with a circumference of 31.4 units?
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Study Notes
Derivatives
- The derivative of a polynomial function can be found using the power rule: ( f'(x) = n \cdot ax^{n-1} ).
- For ( f(x) = 3x^4 - 5x^3 + 2x - 7 ):
- Differentiate each term:
- ( 3x^4 ) becomes ( 12x^3 )
- ( -5x^3 ) becomes ( -15x^2 )
- ( 2x ) becomes ( 2 )
- ( -7 ) (a constant) becomes ( 0 )
- Differentiate each term:
- Thus, ( f'(x) = 12x^3 - 15x^2 + 2 ).
Limits
- To evaluate ( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} ), factor the numerator:
- ( x^2 - 4 = (x - 2)(x + 2) ).
- The expression simplifies to ( \frac{(x - 2)(x + 2)}{(x - 2)} = x + 2 ) for ( x \neq 2 ).
- Computing the limit as ( x ) approaches 2 gives ( 2 + 2 = 4 ).
Geometric Sequences
- A geometric sequence has a first term and a common ratio.
- Given:
- First term ( a = 3 )
- Common ratio ( r = 2 )
- The nth term formula: ( a_n = ar^{n-1} ).
- For the 5th term:
- ( a_5 = 3 \cdot 2^{5-1} = 3 \cdot 16 = 48 ).
Rational Root Theorem
- The Rational Root Theorem suggests potential roots of a polynomial can be the factors of the constant term divided by factors of the leading coefficient.
- For ( 2x^3 - 7x^2 + 4x - 1 ):
- Constant term: -1
- Leading coefficient: 2
- Possible rational roots include ( \pm 1, \pm \frac{1}{2} ).
Area of a Circle
- The circumference ( C ) of a circle is given by ( C = 2\pi r ).
- For a circumference of 31.4 units, set ( 2\pi r = 31.4 ).
- Solving for radius ( r ):
- ( r = \frac{31.4}{2\pi} \approx 5 ).
- Area ( A ) is given by ( A = \pi r^2 ):
- ( A \approx \pi \cdot 5^2 = 25\pi ) square units.
Studying That Suits You
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Description
Test your knowledge in Mathematical Thinking II with this quiz featuring 10 questions on derivatives, limits, sequences, polynomial roots, and geometry. Ensure to prepare well to cover all vital concepts presented in this subject.
