3. (a) (3 points) Write the definition of the derivative of a differentiable function f() at...
3. (a) (3 points) Write the definition of the derivative of a differentiable function f(x) at = a; (b) (7 points) using the definition of derivative as in (a), find the derivative of the function f(x) = Vx at a = 2. (c) EXTRA CREDIT (2 points): State the MEAN VALUE THEOREM (you can also draw a picture) and give its PHYSICAL interpretation in terms of INSTANTANEOUS and AV- ERAGE VELOCITIES.
5x + 1 Use the definition of the derivative to find the derivative of the function f(x) = *-*2. Then find all x-values (if any) where the tangent line is horizontal. If the tangent line is horizontal for all X, write R for your answer. If the tangent line is never horizontal, write None for your answer Answer 3 Points Keypad 11 1'(x) = 2 Tangent is horizontal at x = Prev Nex If f(3) = -1, f(3) = 17,...
7*). Using this definition, Derivative of a function f (x) can be expressed as f'(x) = lim ** find out the first order derivative (f'(x)) of the following functions: h 0 h f(x) = 2x2 + 4 f(x) = 2x (4 points) (4 points)
D1.1. Evaluate f'(a) by using the definition of derivative of a function f(x) = 4x2 + 3x – 5 at a = -2. [4 Marks] D1.2. (a) Find the derivative of y = 4 sin( V1 + Vx). (b) If y = sin(cos(tan(x2 + 3x – 2))), then find the first derivative. [3 Marks] D1.3. Using logarithmic differentiation, find the derivative of y = (sec x)+”.
Using the definition of the derivative, find f'(x) for the following function. 3. f(x) = x2 + x - 1 the definition of the derivative f'(x) = lim f(x+h)-f(x) h h-0 What is the slope of tangent line to this curve at x = -1?
Using the definition, calculate the derivative of the function. Then find the values of the derivative as specified. f(x) = 3 + x2: f'(- 9), F 'O), f (9) Using the definition, calculate the derivative of the function. Then find the values of the derivative as specified. 4 96) = 3. g'(-2), g'(4), g(6) o't)= dx if y = 7x3 dy || s={3 - 4+, t= -6 s'(t)= 0 ne indig y=f(x)= 3 + 14-x, (x,y)= (0,5) The derivative of...
2. Given f(x) = find the derivative using the definition of derivative. 3. Find A and B given that the function, f(x), is continuous at x = 6. V | f(x) = {B (Ar - 42 > 6 4. Find the slope of the tangent line to the curve 2.2 - 2xy + 3y2 = 10 at the point (1,2).
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.