d²r/dt²=3t²i-tj initial conditions dr/dt=-9i and r(0)=8i-10k

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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d²r/dt²=3t²i-tj initial conditions dr/dt=-9i and r(0)=8i-10k

Expert Solution
Step 1: Given data

Given space that
fraction numerator straight d squared straight r over denominator dt squared end fraction equals 3 straight t squared straight i minus tj
Both space side space integrating comma space we space get
integral straight d open parentheses dr over dt close parentheses equals integral left parenthesis 3 straight t squared straight i minus tj right parenthesis dt space space space space space space space space space space space space space space space space space space space space space open parentheses Since space formula space of space the space integration space integral straight x to the power of straight n dx equals fraction numerator straight x to the power of straight n plus 1 end exponent over denominator straight n plus 1 end fraction close parentheses
dr over dt equals integral left parenthesis 3 straight t squared straight i minus tj right parenthesis dt
dr over dt equals fraction numerator 3 straight t cubed straight i over denominator 3 end fraction minus fraction numerator straight t squared straight j over denominator 2 end fraction plus straight C
where space straight C space is space arbitrary space constant.
Since space at space straight t equals 0 comma space dr over dt equals negative 9 straight i space
minus 9 straight i equals 0 minus 0 plus straight C
straight C equals negative 9 straight i
Hence space dr over dt equals fraction numerator 3 straight t cubed straight i over denominator 3 end fraction minus fraction numerator straight t squared straight j over denominator 2 end fraction minus 9 straight i

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