// Numbas version: exam_results_page_options {"name": "Numbastest uke 38: Eksponensiell vekst", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], []], "questions": [{"name": "Vekstfaktor ved \u00f8kning", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

En vare koster $\\var{pris}$ kroner. Prisen stiger med $\\var{p}$ %.

En måte å regne den nye prisen er

Den nye prisen $=\\var{pris} + \\var{p}\\% \\mathrm{\\, av\\, } \\var{pris}=\\var{pris} + \\dfrac{p}{100}\\cdot\\var{pris}=\\var{pris}\\cdot(1 +\\var{pf})=\\var{vf}\\cdot\\var{pris}$

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Vekstfaktor (VF) i dette tilfellet er

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Den nye prisen blir

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VF ved $\\var{p1}$% økning blir

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Ved VF=$\\var{vf1}$ blir det [[0]] % økning

La oss tenke at en vare som koster $\\var{pris}$ kroner stiger i pris med $\\var{p}$ % årlig. Skriv inn et uttrykk for den nye verdien etter $\\var{year}$ år.

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c) $(ab)^{\\var{a}}a^{\\var{b}}b^{\\var{c}}=(a^{\\var{a}}\\cdot b^{\\var{a}})a^{\\var{b}}b^{\\var{c}}=a^{\\var{a}+\\var{b}}\\cdot b^{\\var{a}+\\var{c}}=a^{\\simplify{{a}+{b}}}\\cdot b^{\\simplify{{a}+{c}}}$

d)$(a^{\\var{a}}a^{\\var{b}}b^{\\var{c}})^{\\var{d}}=(a^{\\var{a}+\\var{b}}\\cdot b^{\\var{c}})^{\\var{d}}=(a^{\\simplify{{a}+{b}}}\\cdot b^{\\var{c}})^{\\var{d}}=a^{\\simplify{{a}+{b}}\\cdot\\var{d}}\\cdot b^{\\var{c}\\cdot\\var{d}}=a^{\\simplify{({a}+{b})*{d}}}\\cdot b^{\\simplify{{c}*{d}}}$

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Du har et uttrykk $\\var{k}^{\\var{m}}$. Velg passende begreper

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Velg passende utregninger for følgende potenser

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Forenkle uttrykkene og skriv svar som produkt av to faktorer $(ab)^{\\var{a}}a^{\\var{b}}b^{\\var{c}}$

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Forenkle uttrykkene og skriv svar som produkt av to faktorer $(a^{\\var{a}}a^{\\var{b}}b^{\\var{c}})^{\\var{d}}$

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Forenkle uttrykkene og skriv svar som produkt av to faktorer $(ab)^{\\var{a}}a^{\\var{b}}b^{\\var{c}}$

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Det er gitt fire grafer til fire eksponentialfunksjoner.

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For hvert uttrykk finn en graf.

En vare koster $\\var{pris}$ kroner. Prisen ble satt ned med $\\var{p}$ %.

En måte å regne den nye prisen er

Den nye prisen $=\\var{pris} - \\var{p}\\% \\mathrm{\\, av\\, } \\var{pris}=\\var{pris} - \\dfrac{p}{100}\\cdot\\var{pris}=\\var{pris}\\cdot(1 -\\var{pf})=\\var{vf}\\cdot\\var{pris}$

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Vekstfaktor (VF) i dette tilfellet er

Den nye prisen blir

VF ved $\\var{p1}$% nedgang blir

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Ved VF=$\\var{vf1}$ blir det [[0]] % nedgang

La oss tenke at en vare som koster $\\var{pris}$ kroner avtar i pris med $\\var{p}$ % årlig. Skriv inn et uttrykk for den nye verdien etter $\\var{year}$ år.

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Grafen nedenfor viser utvikling av bilverdien i løpet av flere år etter den var kjøpt

{geogebra_applet('https://www.geogebra.org/m/gzrr3kaj',defs)}

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Den opprinnelige bilverdien i kroner var

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Årlig verditfall i prosent er (vi antar at prisen stiger med samme prosenttall hvert år)

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Skriv inn funksjons uttrykk som viser bilprisen etter $x$ år

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Forenkle følgende uttrykk

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$\\sqrt{(a^{\\var{a}}}\\cdot (\\sqrt{b})^{\\var{b}}=a^{\\frac{\\var{a}}{2}}\\cdot b^{\\frac{\\var{b}}{2}}=a^{\\simplify{{a/2}}}\\cdot b^{{\\simplify{{b/2}}}}

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$\\sqrt{a^{\\var{a}}}\\cdot (\\sqrt{b})^{\\var{b}}$

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Det er gitt fire grafer til fire potensfunksjoner.

For hvert uttrykk finn en graf.

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For hvert uttrykk finn et grunntall

En logaritmisk likning $\\log_x y=z$ kan skrives i følgende eksponentiell form

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Finn følgende logaritmer:

$\\log 10=$[[0]]

$\\ln 1=$[[1]]

$\\log_2 8=$[[2]]

$\\ln e^3=$[[3]]

$\\lg\\frac{1}{100}=$[[4]]

$\\log_3 81=$[[5]]

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Det er gitt fire grafer til fire logaritmiske funksjoner.

For hvert uttrykk finn en graf.

Bruk regneregler for logaritmer

", "advice": "

Bruk følgende regnereglene

$\\log_a(uv)=\\log_a u+\\log_av$

$\\log_a(\\frac{u}{v})=\\log_a u-\\log_av$

$\\log_a(u^r)=r\\cdot\\log_a u$

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$\\log\\var{ab}=\\log\\var{a}+\\log$[[0]]

$\\log\\var{c}=\\log\\var{cd}-\\log$[[1]]

$\\log\\var{c}^{\\var{a}}=$[[2]]$\\cdot\\log$[[3]]

$\\log\\dfrac{\\var{z}x^{\\var{a}}y^{\\var{b}}}{x^{\\var{c}}y^{-\\var{d}}}=$[[0]]$\\cdot\\log x$+[[1]]$\\cdot\\log y\\, +$[[2]]

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