Practice Questions
- Differential Equations: If \[ \frac{dy}{dx} = (2x - 1)y \], and \( y(1) = e \), find \( y(2) \).
- A) \( 3e \)
- B) \( 3^{e} \)
- C) \( \dfrac{e}{3} \)
- D) \( \dfrac{3}{e} \)
- E) \( e^{3} \)
- Particle Motion: A particle moves along the x-axis with velocity \( v(t) = 2t^{2} - t + 1 \). When \( t = 0 \), the particle is at \( x = -3 \). Find the particle's position at \( t = 2 \).
- A) \( 7 \)
- B) \( \dfrac{7}{3} \)
- C) \( 1 \)
- D) \( 22 \)
- E) \( \dfrac{3}{7} \)
- Partial Fractions Integration: Evaluate the integral:
\[ \int_{3}^{4} \frac{5x + 5}{x^{2} + x - 6} \, dx \]
- A) \( 0 \)
- B) \( \ln\left(\dfrac{7}{3}\right) + \ln(2) \)
- C) \( 2\ln\left(\dfrac{7}{3}\right) - \ln(2) \)
- D) \( 2\ln\left(\dfrac{7}{3}\right) + \ln(2) \)
- E) \( 3\ln\left(\dfrac{7}{3}\right) \)
- Integration Techniques: Evaluate the integral:
\[ \int \frac{1}{x(1 + (\ln x)^{2})} \, dx \]
- A) \( \arctan(\ln x) + C \)
- B) \( \dfrac{1}{2}x^{2}(\ln^{2}(x) \ln(x)) + \dfrac{3}{4}x^{2} + C \)
- C) \( \left[ \dfrac{1}{2}x^{2}(\ln^{2}(x) \ln(x)) + \dfrac{3}{4}x^{2} \right]^{-1} + C \)
- D) \( \arcsin(\ln x) + C \)
- E) \( \dfrac{1}{\arctan(\ln x)} + C \)
- Inverse Functions: If \( f(x) = 2 + \ln(x + 3) \), then its inverse \( f^{-1}(x) \) is:
- A) \( f^{-1}(x) = (2 + \ln(x + 3))^{-1} \)
- B) \( f^{-1}(x) = \ln(x - 2) \)
- C) \( f^{-1}(x) = -(2 + \ln(x + 3))^{-1} \)
- D) \( f^{-1}(x) = e^{x - 2} - 3 \)
- E) \( f^{-1}(x) = e^{x + 2} - 3 \)
- Curve Characteristics: If \( f'(x) < 0 \) and \( f''(x) < 0 \) for all \( x \), which graph represents \( f \)?
- Polynomial Intercepts: The set of all \( K \) such that \( f(x) = x^{4} - 14x^{2} + 24x + K \) has two distinct x-intercepts is:
- A) \( (0, \infty) \)
- B) \( (-\infty, \infty) \)
- C) \( (-100, -11) \cup (-11, 130) \)
- D) \( (-8, 127) \)
- E) \( (-\infty, -11) \cup (-8, 117) \)
- Symmetry & Derivatives: If \( f(-x) = f(x) \) and \( f \) is differentiable for all \( x \), which must be true?
- A) \( f'(-x) = \dfrac{1}{f'(x)} \)
- B) \( f'(-x) = f'(x) \)
- C) \( f'(-x) = -f'(x) \)
- D) \( f'(-x) = -\dfrac{1}{f'(x)} \)
- E) \( f'(-x) = (f'(x))^{-1} \)
- Logarithmic Differentiation: If \( y = x^{x + 1} \), then \( y' = \)
- A) \( (x + 1)x^{x} \)
- B) \( (x + 1)x^{x - 1} \)
- C) \( x \ln(x^{x}) + x + 1 \)
- D) \( x^{x}(x \ln x + x + 1) \)
- E) \( x^{x} \cdot x \ln x \)
- Parametric Derivatives: If \( x = \ln(t + 1) \) and \( y = \ln(t + 2) \), then \( \dfrac{dy}{dx} = \)
- A) \( 1 \)
- B) \( \dfrac{t + 1}{t + 2} \)
- C) \( \dfrac{t + 2}{t + 1} \)
- D) \( \dfrac{1}{(t + 2)\ln(t + 1)} - \dfrac{\ln(t + 2)}{(t + 1)\ln^{2}(t + 1)} \)
- E) \( \dfrac{1}{t + 2} \)
- Limits & Trigonometry: Evaluate:
\[ \lim_{x \to 0} \frac{\sin(x)-\sin(2x)}{x} \]
- A) \( 1 \)
- B) \( 2 \)
- C) \( 0 \)
- D) \( -2 \)
- E) \( -1 \)
- Chain Rule Iteration: If \( y = \sin(\sin(\sin(x))) \), then \( \dfrac{dy}{dx} = \)
- A) \( \cos(\sin(\sin(x))) \)
- B) \( \cos(\cos(\sin(x))) \)
- C) \( \cos(x) \cos(\sin(x)) \cos(\sin(\sin(x))) \)
- D) \( \cos(\cos(\cos(x))) \)
- E) \( \cos(x) \cos(\sin(x)) \cos(\cos(\sin(x))) \)
- Implicit Differentiation: Find \( \dfrac{dy}{dx} \) if \( x = \ln(y - e^{-y}) \).
- A) \( y + e^{-y} \)
- B) \( \dfrac{y - e^{-y}}{1 - e^{-y}} \)
- C) \( \dfrac{y + e^{-y}}{1 - e^{-y}} \)
- D) \( \dfrac{y - e^{-y}}{1 + e^{-y}} \)
- E) \( y - e^{-y} \)
- Parametric Derivatives: Find \( \dfrac{dy}{dx} \) for \( x = \ln t + t \) and \( y = t - \ln t \).
- A) \( \dfrac{t - 1}{t + 1} \)
- B) \( \dfrac{t + 1}{t - 1} \)
- C) \( -\dfrac{t - 1}{t + 1} \)
- D) \( -\dfrac{t + 1}{t - 1} \)
- E) \( 1 \)
- Composite Functions: Given \( g(x) = f(h(x)) \), \( f(0) = 1 \), \( h(0) = 2 \), \( g'(0) = 3 \), \( f'(0) = 7 \), \( f'(2) = 6 \), and \( f'(5) = 8 \), find \( h'(0) \).
- A) \( 2 \)
- B) \( \dfrac{1}{2} \)
- C) \( -\dfrac{1}{2} \)
- D) \( 1 \)
- E) \( -1 \)
- Function Composition: If \( f(g(x)) = 2x \) and \( f(t) = e^{2t + 1} \), then \( g(x) = \)
- A) \( \ln(2x) \)
- B) \( e^{2x} \)
- C) \( \ln(2x) + 1 \)
- D) \( \dfrac{\ln(2x) - 1}{2} \)
- E) \( \ln(x) \)
- Area under Curves: Half a period of \( y = \sin x \) from \( 0 \) to \( \pi \) is split into two equal areas by a line through the origin intersecting the curve at \( x = K \). Which equation does \( K \) satisfy?
- A) \( \sin K = K \)
- B) \( \cos K = K \sin K \)
- C) \( K \cos K = \sin K \)
- D) \( (2 + K)\sin K + 2\cos K = 0 \)
- E) \( K \sin K + 2\cos K = 0 \)
- Polar Curves: Which graph represents the polar curve \( r = -\csc t \)?
- Absolute Value Derivatives: Find \( \dfrac{dy}{dx} \) if \( y = |x^{2} + 2x - 1| \).
- A) \( \dfrac{(2x + 2)(x^{2} + 2x - 1)}{|x^{2} + 2x - 1|} \)
- B) \( |2x + 2| \)
- C) \( -|2x + 2| \)
- D) \( |2x - 2| \)
- E) \( -2x + 2 \)
- Polynomial Analysis: For \( f(x) = x^{5} + 5x^{4} - 40x^{2} - 80x - 48 \), which statement is true?
- A) It has no x-intercepts
- B) It has 3 inflection points
- C) It is concave up for all \( x \)
- D) It has 1 inflection point
- E) It has 2 inflection points