a. Graph the function. b. Draw tangent lines to the graph at point whose x -coordinates are –2, 0, and 1. c. Find f ' ( x ) by determining lim x → 0 f ( x + h ) − f ( x ) h . d. d) Find f ' ( − 2 ) , f ' ( 0 , ) and f ' ( 1 ) . These slopes should match those of the lines you drew in part ( b ). f ( x ) = − x 3
a. Graph the function. b. Draw tangent lines to the graph at point whose x -coordinates are –2, 0, and 1. c. Find f ' ( x ) by determining lim x → 0 f ( x + h ) − f ( x ) h . d. d) Find f ' ( − 2 ) , f ' ( 0 , ) and f ' ( 1 ) . These slopes should match those of the lines you drew in part ( b ). f ( x ) = − x 3
Solution Summary: The author illustrates the graph of the function f(x)=-x-3.
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.
The graph of the function f(x) is shown below. It consists of a semicircle and two
straight line segments.
150
+
-3
-2
y 3
2
-1
1
2
(b) Which of the following numbers is the smallest:
ƒ' (−3.95), ƒ' (−1.5), ƒ' (−0.05), ƒ' (1), ƒ' (3.05)?
3
5
X
9. For each of the following functions:
a)Find an equation of the tangent line to the curve at the given point and
b) find the values of x for which the tangent line is horizontal.
i) f(x)=x* - x² at the point (1, 0)
ii) g(x) =
at the point (2, -1/2)
iii) w(x) =
1
at x=-1
1+x?
iv) s(t) = tan t t=0 and t= T
4.
%3D
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY