MTT200 Calculus II Problem Set 2 PDF

MTT200 Winter 2024-2025 Chapter 7: Problem Set # 2 Calculus II 13 January Check your understanding: Find the derivative of the following functions: 1. y = 4x 2. f (x) = 4−x 3. y = 4sin x 4. y = ln 2x d 5. dx ln(x2 + 3) 6. f (θ) = ln(...

MTT200 Winter 2024-2025 Chapter 7: Problem Set # 2 Calculus II 13 January Check your understanding: Find the derivative of the following functions: 1. y = 4x 2. f (x) = 4−x 3. y = 4sin x 4. y = ln 2x d 5. dx ln(x2 + 3) 6. f (θ) = ln(sec(θ) + tan(θ)) Try: 1 Try: q (2x+1)10 Find the derivative of y = (x−3)5. 2 Inverse Trigonometric Functions: Examples: 1. Find d dx (sin−1 x2 ) 2. Find d dx (sec−1 5x4 ) 3 4 5 1. Evaluate the definite integral Z 1 2 dx √ 0 1 − x2 2. Find Z dx 9 + x2 3. Evaluate the integral Z dx √ 9 − x2 4. Evaluate the integral Z dx 25 + 4x2 5. Evaluate the integral Z dx √ 4 − 9x2 6. Evaluate the integral Z dx √ 1 − 16x2 7. Evaluate the integral 4−x Z √ dx 16 − x2 6 7 8 Derivatives of Hyperbolic Functions: Integrals of Hyperbolic Functions: 9  d √ 1. Find dt tanh 1 + t2 2. Find Z coth 5x dx 3. Evaluate Z 1 sinh2 x dx 0 4. Evaluate Z ln 2 4ex sinh x dx 0 10 11 12 Derivatives of Inverse Hyperbolic Functions: Integrals of Inverse Hyperbolic Functions: 13 1. Evaluate Z 1 2dx √ 0 3 + 4x2 14 15 16 Relative Rates of Growth: EXAMPLE: Let’s compare the growth rates of several common functions. 1. y = ex and y = x2 as x −→ ∞. 2. y = 2x and y = x as x −→ ∞. 1 3. y = ln x and y = x n , for any positive integer n, as x −→ ∞. 4. y = 3x and y = 2x as x −→ ∞. 5. y = x2 and y = ln x as x −→ ∞. 17  6. Show that ln x = o(x) as x −→ ∞. 7. Show that x2 = o(x3 + 1) as x −→ ∞. 8. Show that x + sin x = O(x) as x −→ ∞. 9. Show that ex + x2 = O(ex ) as x −→ ∞. 10. Show that x = O(ex ) as x −→ ∞. 18 19