Soru 1 (Limits & L'Hopital's Rule)
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Evaluate lim (x→0) (sin(5x) / x).
💡 Adım Adım Çözüm:
Using L'Hopital's rule or standard limit lim (u→0) sin(u)/u = 1: lim 5 * (sin(5x)/5x) = 5 * 1 = 5.
Soru 2 (Derivatives (Chain Rule))
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Find the derivative d/dx [e^(x² + 3x)].
💡 Adım Adım Çözüm:
By Chain Rule: d/dx [e^u] = u' * e^u = (2x + 3) * e^(x² + 3x).
Soru 3 (Definite Integrals)
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Evaluate ∫ (from 1 to e) (1/x) dx.
💡 Adım Adım Çözüm:
Antiderivative of 1/x is ln|x|. [ln(x)] from 1 to e = ln(e) - ln(1) = 1 - 0 = 1.
Soru 4 (Integration by Parts)
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Evaluate ∫ x * cos(x) dx.
💡 Adım Adım Çözüm:
Let u = x, dv = cos(x)dx => du = dx, v = sin(x). ∫ u dv = u v - ∫ v du = x sin(x) - ∫ sin(x)dx = x sin(x) + cos(x) + C.
Soru 5 (Implicit Differentiation)
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For the curve x² + y² = 25, find dy/dx at the point (3, 4).
💡 Adım Adım Çözüm:
Differentiate both sides: 2x + 2y(dy/dx) = 0 => dy/dx = -x/y. At (3, 4), dy/dx = -3/4.
Soru 6 (Taylor & Maclaurin Series)
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What is the Maclaurin series expansion of cos(x)?
💡 Adım Adım Çözüm:
The Maclaurin series for cos(x) is 1 - x²/2! + x⁴/4! - x⁶/6! + ... = ∑ (-1)^n x^(2n) / (2n)!.
Soru 7 (Partial Derivatives)
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For f(x, y) = x³y² + 4x²y, find the partial derivative ∂f/∂x.
💡 Adım Adım Çözüm:
Treat y as constant: ∂f/∂x = 3x² * y² + 8x * y = 3x²y² + 8xy.
Soru 8 (Differential Equations)
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Solve the differential equation dy/dx = 2xy with y(0) = 3.
💡 Adım Adım Çözüm:
Separating variables: (1/y)dy = 2x dx => ln|y| = x² + C => y = C e^(x²). Since y(0) = 3, C = 3 => y = 3e^(x²).
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