Classroom notes …x ¡ 1†…x ‡ 2† 50 …¡x ¡ 3†…x ‡ 4†
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to find the domain of f …x†
Example 8. If f …x† ˆ …1=3†x3 ¡ 6x2 ‡ 11x ¡ 6, ®nd the intervals over which … † f x is below the x-axis, the intervals over which f …x† is increasing, and the interval over which f …x† is concave down. Hints: (1) For below the x-axis, set f …x† < 0 and solve for x to get the intervals. 0 (2) For increasing, set f …x† > 0 and solve for x to get the intervals. 00 (3) For concave down, set f …x† < 0 and solve for x to get the interval. Example 9. Solve for x if ‰x2 ¡ x ¡ 6Š > 5, where ‰x2 ¡ x ¡ 6Š is a greatest integer function of x. Hint: We can solve x2 ¡ x ¡ 6 5 6 to ®nd the solution. Acknowledgment The author thanks the referees for their helpful suggestions.
A graphic study of the properties of real-valued functions of two variables F. MARTIÂNEZ* and C. VINUESA Departamento de MatemaÂticas, Universidad de CaÂdiz, Duque de NaÂjera 8, 11002 CaÂdiz, Spain; e-mail:
[email protected];
[email protected] (Received 13 June 2001 ) This note presents some simple examples illustrating the di erences between the concepts of continuity, existence of partial derivatives and di erentiability.
In the study of the functions f : R 2 ! R , the concepts of continuity, partial derivatives, directional derivatives and di erentiability, and the connection among these concepts are very important. However, these connections are di cult to understand by ®rst-year students of calculus. It is common, for instance, for them to think that the existence of partial derivatives fx …a; b† and fy …a; b† implies that the plane z ¡ f …a; b† ˆ fx …a; b†…x ¡ a† ‡ fy …a; b†…y ¡ b† is a tangent plane to the surface z ˆ f …x; y† at …a; b; f …a; b†† and so f is di erentiable at …a; b†. The purpose of this note is to present simple, clarifying and easy-to-draw examples to illustrate the di erences among the above concepts. In this way, we show that there is no relation between continuity and the existence of partial derivatives at a point for these functions (®gures 1 and 2): * Author to whom correspondence should be addressed.
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Classroom notes
Figure 1
Figure 2
Table 1
f …x; y† ˆ
»
1 0
if y ˆ x; x 6ˆ 0 otherwise
8 0 such that the box Q0 ˆ …x…t0 † ¡ r0 ; x…t0 † ‡ r0 † £ …y…t0 † ¡ r0 ; y…t0 † ‡ r0 † £ …z…t0 † ¡ r0 ; z…t0 † ‡ r0 †, veri®es: \ Q0 S»C Then:
(1) If C is included in a plane in the form x ˆ k, k 2 R then fx is not continuous at …a; b†: (2) If C is included in a plane in the form y ˆ h, h 2 R then fy is not continuous at …a; b†: (3) If the intersection between C and any plane x ˆ k; k 2 R and any plane y ˆ h, h 2 R is at the most a point then neither fx nor fy are continuous at …a; b†. Proof. (1) We can suppose that C is included in the plane x ˆ 0: We take t0 2 …t1 ; t2 †, then x…t0 † ˆ 0 and the intersection between S with the plane y ˆ y…t0 † is: » z…t0 † if x ˆ x…t0 † ˆ 0 z ˆ g…x † ˆ f …x; y…t0 †† if x 6ˆ x…t0 † ˆ 0
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Classroom notes
Figure 8
Figure 9
By (ii) we have that lim x!0 g…x† ˆ limx!0 f …x; y…t0 †† 6ˆ z…t0 †, then g is not derivable at x…t0 † ˆ 0 and does not exist fx …0; y…t0 †† for each t0 2 …t1 ; t2 †. Finally, by …i† we have that fx is not continuous at …a; b†. & In a similar way can be proved (2) and (3). Finally, we give the examples for table 2. …b† 6) …6†. Let f be the function: » 1 f …x; y† ˆ 0
if x ˆ 0 or y ˆ 0 otherwise
The function f (®gure 8) is not continuous at …0; 0† and fx …0; 0† ˆ fy …0; 0† ˆ 0 but the other directional derivatives do not exist at …0; 0†: …6† 6) …5†. Let f be the function: f …x; y† ˆ
8
