1. (20 points) Find derivatives of the following functions. (a) f(x) = 1012 (b) g(x) =...
Find the derivatives of the following functions. a) f(x) = sin(x²) b) g(x) = ln(cos(x)) c) h(z) – In(1002? + 47x)? d) k(z) = resin(2)
CALCULUS EXERCISES 1. (10 points) Derivatives. Calculate the following derivatives: (a) f(x) = 67 (b) f(x) 32 +8x? (c) f(x) = (6x ) (d) f(x) = (e) fix,y) Inix + 2) + 2xy + 2. (20 points) Optimization. Check if the following functions have a global optimum, and if so, whether is a minimum or maximum. (a) f(x)=-20-42+4x (b) f(x) 100+62-8x 3. (10 points) More optimization. Explain the intuition behind the first-order condition of an opti- mization problem. APPLICATIONS
Find the derivatives of the following functions (A) y=xt-x2 (B) y=x2 (C) y-3x3 + 4x -3 (D) y=x (F) f(x)- 100x100 (G) fx)- (H) f(x)=- 1-5r ) f(x)= (J f(x) - (x + 1)(x3 +5x + 2) (Requires quotient rule) 4+x For f(x) = 6x3, find f(x) and f"(x), and fm(x). For f(x)- 5x3 -3x2+x-20, find f(x), f"(x), and f""(x)
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem) Question 2...
Find the derivative of the following functions: Inx 22 C. a. F(x) = In (3x) b. Y=: Y=52x-1 d. Y= log1032 e. Y=In(x2-2)2/3 f. Y= In g. Y= log2 (2x - 1) h. Y= 8** e 1+e+
Find the second and third derivatives of the following functions by using Matlab 1. f(x) = 3cosx-2sinx x² 2. f(x) = ln(x) - 1 3. f(x) = 5x3 - 2x2 + 7x + 10
1. (based on exercise 6.5 on page 301) For each of the following functions, find their first and second derivatives, and use these to find the function's critical points. Characterize each critical point as a local minimum, maximum, saddle point, or something else. (a) f(x, y-x2-4ry + y2 (b) f(x,y)=x4-4xy+94. (c) f(x, y) 2x3-3z2-62y(x-y-1). (d) f(x, y) = y4-v2 + 2y(1-x) + 1. (e) For the function in item 1d, what is the steepest descent direction at (x, y) (0,0)?...
Question 2. (20 points, 4 each) Apply the basic differentiation rules to find the derivatives of the functions given below. (a) f(x)=In(5x2+3x) - 2Vx3 (b) f(x)=(x2-3) (e3x-1 +2x) (c) 2x4+7xy+5x3=120 (d) S¢ + x)dx (e) S(x2 + e2x)dx
Evaluate the following f(x)=x2-1 and g(x) = 3x +5. :a. f(-3) b. g(-2) c. f(0) d. g(5) 2. Find the x and y intercepts of the following functions: a) f(x) = x2 - 5x + 6 = 0b) h(x) = -2x + 20
3. For each of the following functions, (i) Determine the domain, (ii) Find their first derivatives, (iii) Find their second derivatives, (iv) Determine whether they are globally concave? Why or why not? (a) f(x) = 4x? - (b) f(x) = ln(22 - 2) (c) f(x) = e