Use the lagrange-multiplier method to find the stationary

Question

Exercise 10.1

1. Taking for granted that e^t is its own derivative, use the chain rule to find dy/dt for the following:

(a) y = 4e^3t

Exercise 10.2

1. What is the instantaneous rate of growth of y in each of the following?

a) y = e^0.07t  (b) y = 15e^0.03t (c) = Ae^0.4t (d) y = 0.03e^t

Exercise 10.3

1. What are the values of the following logarithms?

(a) log₁₀ 0.0001

2. Evaluate the following:

(a) log_e e^-4 (b) log_e(1/e²) (c) lne^x - e^lnx

Exercise 10.5

1. Find the derivatives of:

(a) y = e^{1 - 9t} (b) y = 5e^{2- t2} (c) y = xe^x (d) y = x²e^2x (e) y = axe^{bx +c}

2. Find the derivatives of:

(a) y = ln (at^c) (b) y = 5 ln(t +1)² (c) y = ln [x(1 - x)⁸] (d) y = ln(2x/(1+x)) (h) y = 5x⁴ In x² 7t

3. Find the derivatives of the following by first taking the natural log of both side

(a) y = (x² + 3)e^x2+1

Exercise 12.2

1. Use the Lagrange-multiplier method to find the stationary values of z:

(a) z = X(Y + 4), subject to x + y = 8.

(b) z = 7 - y+ x², subject to x + y = 0.

2. Write the Lagrangian function and the first-order condition for stationary values (without solving the equations) for each of the following:

(a) z= x + 2y+ 3w + xy- yw, subject to x + y+ 2w = 10.

(b) z = x² + 2xy+ yw², subject to 2x + y + w² = 24 and x + w = 8.