Browse Questions

# Using differentials, find the approximate value of each of the following up to 3 places of decimal. $(xiii)\;(81.5)^{\Large\frac{1}{4}}$

This is thirteenth part of multipart q1

Toolbox:
• Let $y=f(x)$
• $\Delta x$ denote a small increment in $x$
• $\Delta y=f(x+\Delta x)-f(x)$
• $dy=\big(\large\frac{dy}{dx}\big)\Delta x$
Step 1:
$y=x^{\Large\frac{1}{4}}$
$x=81$
$\Delta x=0.5$
$\Delta y=(x+\Delta x)^{\Large\frac{1}{4}}-x^{\Large\frac{1}{4}}$
$\quad\;\;=(81.5)^{\Large\frac{1}{4}}-(81)^{\Large\frac{1}{4}}$
$\quad\;\;=(81.5)^{\Large\frac{1}{4}}-3$
$(81.5)^{\Large\frac{1}{4}}=3+\Delta y$
Step 2:
$dy$ is approximately equal to $\Delta y$
$dy=\large\frac{dy}{dx}$$\Delta x \quad=\large\frac{1}{4}x^{-\Large\frac{3}{4}}$$.\Delta x$
$\quad=\large\frac{1}{4x^{\Large\frac{3}{4}}}$$.\Delta x \quad=\large\frac{1}{4(81)^{\Large\frac{3}{4}}}$$\times 0.5$
$\quad=\large\frac{0.5}{4\times 27}$
$\quad=\large\frac{0.5}{108}$
$\quad=0.0046296$
Step 3:
$(81.5)^{\Large\frac{1}{4}}=3+0.0046296$
$\qquad\;\;\;\;\;=3.0046296$
$\qquad\;\;\;\;\;=3.005$
{"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": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9834991693496704, "perplexity": 4462.768204709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542712.12/warc/CC-MAIN-20161202170902-00476-ip-10-31-129-80.ec2.internal.warc.gz"}