201-NYA-05-2023-3-annotated-plus-inverse-trig

Page 1: Limits and Function Analysis

Limits Evaluation

• Evaluate the limits:

  • (a) ( \lim_{x \to 1} \frac{x^2 - 1}{x^2 + 3x + 2} )

  • (b) ( \lim_{x \to 1^+} \cos\left(\frac{\pi x}{3}\right) \ln(x) )

  • (c) ( \lim_{x \to 1} \frac{x - \sqrt{x^2 - x + 7}}{3x + 7} )

  • (d) ( \lim_{x \to 0} \frac{x^2}{x \tan(x)} )

  • (e) ( \lim_{x \to 0} \frac{1}{|2x - 1|} \cdot \frac{1}{|2x + 1|} )

Function Analysis

• Consider the function: ( f(x) = 3e^x + 4e^{-x} )

  • (a) Domain of ( f )

  • (b) Horizontal asymptotes of ( f )

  • (c) Vertical asymptote of ( f )

  • (d) Identify the correct graph of ( f )

Page 2: Continuity and Derivatives

Continuity

• Find the value of ( a ) for continuity at ( x = 0 ):

  • ( f(x) = \begin{cases}    a \sin(ax) + x, & x < 0 \    2, & x = 0 \    5e^x + 2a - a^2, & x > 0 \ \end{cases} )

Derivatives

• Use limit definition to find ( f'(x) ) for ( f(x) = \sqrt[5]{3x} )

Implicit Differentiation

• Find ( \frac{dy}{dx} ) for:

  • (a) ( y = \sqrt[5]{x^2 + 2x + \log_2(x)} e^2 )

  • (b) ( y = \tan^3(x^2) \sec(x) )

  • (c) ( y = e^{2x} + \sin(x) \sqrt{x} + \sqrt{x} )

  • (d) ( y = (x + 1)\cos(x) )

  • (e) ( e^{xy} = (x + y)^5 )

Logarithmic Differentiation

• Find the derivative using logarithmic differentiation for ( y = 4\sqrt{x^3 + 1} 7x \tan(x) )

Geometry in Calculus

• Find points on the ellipse ( 2x^2 + y^2 = 22 ) where tangent lines are parallel to ( y = 3x + 8 )

Page 3: Critical Numbers and Extrema

Critical Points

• Find critical numbers for ( f(x) = 3(2x - 1)^{1/3}(x - 2)^{2/3} )

Domain & Behavior

• For ( f(x) = x^2 - 2 \ln(x) ):

  • (a) Domain of the function

  • (b) Intervals of concavity

Optimization

• Absolute maximum and minimum for ( f(x) = \ln(4x^4 + x^2 + 1) ) on ([1, 1])

Function Analysis

• Consider function and derivatives:

  • ( f(x) = 3 - 2\sqrt[3]{x^2} - x arr ext{Calculate derivatives} )

  • (a) Domain and intercepts

  • (b) Asymptotes

  • (c) Intervals of increase or decrease

  • (d) Local extrema

  • (e) Upward and downward concavity

  • (f) Inflection points

Volume Optimization

• Find largest possible volume of a right circular cylinder inscribed in a sphere of radius ( \sqrt{3} ) (use ( V = \pi r^2 h ))

Page 4: Integration

Integral Evaluation

• Evaluate the integrals:

  • (a) ( \int_1^4 \frac{x^2 + \sqrt{x}}{x} , dx )

  • (b) ( \int \left(3 - \sqrt[3]{x^2} + e + \frac{1}{x}\right) , dx )

  • (c) ( \int \left( \sec^2(x) + \csc^3(x) \sin(x) \right) , dx )

Position Function

• Find position function ( s(t) ) given acceleration ( a(t) = 3 \sin(t) + \cos(t) ); ( s(0) = 4, s(\pi) = 4 )

Function Derivation

• For ( f(x) = \int_2^x \frac{\sin(t)}{t + \sin(t)} , dt ):

  • (a) Find ( f(2) ).

  • (b) Find ( f'(x) ).

  • (c) Show horizontal tangent at ( x = \pi )

Limit Evaluation

• Evaluate ( \lim_{n \to \infty} \sum_{i=1}^n \frac{e^{i/n}}{n} ) as a definite integral over ([0, 1]).

Page 5: Final Answers and Review

Summary of Answers

1. Limit Answers:

• a) 2, b) 1, c) ( \frac{2}{3} ), d) 1, e) 4

2. Function Analysis:

• a) Domain: ( \mathbb{R} \setminus {\ln(2)} )

• b) Horizontal at ( y=3, y=2 )

• c) Vertical at ( x = \ln(2) )

3. Critical Points:

• a) ( a = 1 )

4. Derivative Results:

• a) ( f'(x) = \frac{3}{2\sqrt[5]{3} - 3x} )

5. Miscellaneous Derivatives and Values

6. Maximum Volume: ( V_{max} = 4\pi )

7. Critical Numbers: ( x = \frac{1}{2}, 1, 2 )

8. Domain findings and concavity for functions。

9. Evaluate integrals with results.

10. Summary of positional changes and functions over intervals.