Demonstrate an understanding the concept of differential

Question

- Demonstrate an understanding the concept of differential and Integral Calculus;

- Demonstrate an understanding how to calculate and solve engineering problem using differential and Integral Calculus.

Question: 1 Differentiate the following with respect to the variable:

1) y = 5 cos2x+11
2) y = e^0.5x-1
3) y = 4 ln t/2+10
4) y = e^x/x²
5) y = 2√(x²-1)
6) y = 2x³- 4/x³ + 4√(x³ ) + 60
7) f(θ) = 2sin?(4θ+1) - cos?(3θ-1)

Question: 2 Locate the turning point on the curve y=3x² - 6x + 1. And determine whether it is a maximum or minimum.

Question: 3 Determine:

1) ∫√x dx

2) ∫(1-x)² dx

3) 4∫?x dx

4) ₁∫²x² dx

5) ∫ tanx.sec²x dx

Question: 4 Determine :

1) ∫(2x-1)⁸ dx

2) ∫ lnθ/θ dθ

Question: 5 Determine ₄∫⁵12/(x²-9) dx

Question: 6 Using the method of integration by parts, determine

1) ∫ x e^x dx
2) ₁∫² lnx dx

Question: 7  A spherical balloon is being blown up such that its volume increases at the constant rate of 2.00 m³/min. Find the rate (m/min) at which the radius is increasing when it is 1.00 m.

Question: 8 Show that ∫dx/(x² + a²) = 1/a tan^-1 x/a + C by using the substitution x = a tanθ.