beginners in calculus

i am using the Stewart calculus early transcendentals text and in chapter $2.4$there is a question:

use the given graph of f to find the number delta such that

if $0<|x-5|< \delta$ then $|f(x) -3|< 0.6$

Is the answer: right delta = 0.7 and left delta = 1.0? If so pick the smallest ,correct.....

Any help would be appreciated.

-

You just use the provided graph: Draw a vertical strip, vertically  centered at $y=3$, of height $1.2$.  Now draw a horizontal strip centered about $x=5$. What width would this strip have to have in order for the graph of $f$, excluding the point $(5,f(5))$ possibly, to be contained in the rectangle formed by the vertical and horizontal strips? Find the maximum such  value using the provided graph. $\delta$ will be half of the width found. – David Mitra Feb 3 at 17:25
If they give you $f(x)$ by a graph, presumably you should estimate the points where the condition just starts to fail by eye... – vonbrand Feb 3 at 17:25

2 Answers

How far can you get away from $x=5$ before the function's value exceeds 3.6 or is below 2.4?

-

so right delta = 0.7 and left delta = 1.0? – codenamejupiterx Feb 3 at 21:50
I can't say without seeing the graph, but, you are correct (as you stated in your edit to the question) that you should pick the least of the two. – Jonathan Feb 4 at 0:36
Thank You for your help! – codenamejupiterx Feb 4 at 6:58

Maybe this will help ?

You want something in the nature of :

5 - delta < x < 5 + delta

and 3-0.6 = 2.4 < f(x) < 3 + 0.6 = 3.6

If they gave you the graph for f(x), I think that implies you can estimate the bounds (thus lead to the finding of delta) by some eye-balling (Since you already know the lower & upper bound for f(x), using the 2nd condition that I listed above). Using the strategy as in David's comment also works.

-

so right delta = 0.7 and left delta = 1.0? – codenamejupiterx Feb 3 at 21:58
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8981708884239197, "perplexity_flag": "middle"}