In parts (a)-(c), sketch a continuous curve y = f x with the stated properties. (a) f 2 = 4 , f ′ 2 = 0 , f ″ x > 0 for all x (b) f 2 = 4 , f ′ 2 = 0 , f ″ x < 0 for x < 2 , f ″ x > 0 for x > 2 (c) f 2 = 4 , f ″ x < 0 for x ≠ 2 and lim x → 2 + f ′ x = + ∞ , lim x → 2 − f ′ x = − ∞
In parts (a)-(c), sketch a continuous curve y = f x with the stated properties. (a) f 2 = 4 , f ′ 2 = 0 , f ″ x > 0 for all x (b) f 2 = 4 , f ′ 2 = 0 , f ″ x < 0 for x < 2 , f ″ x > 0 for x > 2 (c) f 2 = 4 , f ″ x < 0 for x ≠ 2 and lim x → 2 + f ′ x = + ∞ , lim x → 2 − f ′ x = − ∞
Prob. 6 (a) (10 point) Let f(x) = 2x² – 3. Find ƒ'(−2) using only the limit definition of
derivatives.
(b) (10 p.) If ƒ(x) = √√x + 6, find the derivative f'(c) at an arbitrary point c using only the
limit definition of derivatives.
Estimate of f′(2):
Consider the three points (2, 2), (4,4), and (6, —2).
(a) Supposed that at (2, 2), we know that fx = fy = 0 and ƒxx 0,
function near the point (4,4)? ?
fxy
fry = 0. What can we conclude about the behavior of this
(c) Supposed that at (6, -2), we know that fx = fy = 0 and fxx > 0, fyy < 0, and fxy = 0. What can we conclude about the behavior of
this function near the point (6, -2)? ?
Using this information, on a separate sheet of paper sketch a possible contour diagram for f.
Precalculus: Mathematics for Calculus - 6th Edition
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY