document.write( "Question 1204791: Let f'''(t) = -10t - 10 sqrt(t). I found f''(t) = -5t^2 - 20/3t^(3/2) + C but I am struggling to find f'(t) and f(t). I had found an equation for f'(t) = -5/3t^3 - 8/3t^(5/2) + D but the computer says it's not correct. \n" ); document.write( "
Algebra.Com's Answer #841286 by math_tutor2020(3817)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "The tickmark notation is very limited, so I'll use exponents to represent the derivative level.
\n" ); document.write( "f^3 = 3rd derivative
\n" ); document.write( "f^2 = 2nd derivative
\n" ); document.write( "f^1 = 1st derivative
\n" ); document.write( "f = original function\r
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\n" ); document.write( "\n" ); document.write( "f^3 = -10t - 10*sqrt(t)
\n" ); document.write( "f^3 = -10t - 10*t^(1/2)
\n" ); document.write( "f^2 = integral[ f^3 ]
\n" ); document.write( "f^2 = integral[ -10t - 10*t^(1/2) ]
\n" ); document.write( "f^2 = -10*(1/(1+1))*t^(1+1) - 10*(1/(0.5+1))*t^(1/2+1) + C
\n" ); document.write( "f^2 = -5t^2 - (20/3)t^(3/2) + C\r
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\n" ); document.write( "\n" ); document.write( "As a check,
\n" ); document.write( "f^3 = derivative[ f^2 ]
\n" ); document.write( "f^3 = derivative[ -5t^2 - (20/3)t^(3/2) + C ]
\n" ); document.write( "f^3 = derivative[ -5t^2 ] - derivative[ (20/3)t^(3/2) ] + derivative[ C ]
\n" ); document.write( "f^3 = 2*(-5)t^(2-1) - (3/2)*(20/3)*t^(3/2-1) + 0
\n" ); document.write( "f^3 = -10t - 10*t^(1/2)
\n" ); document.write( "f^3 = -10t - 10*sqrt(t)\r
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\n" ); document.write( "\n" ); document.write( "The CAS (computer algebra system) feature of GeoGebra can be used to verify this.
\n" ); document.write( "WolframAlpha is another good option.
\n" ); document.write( "Feel free to explore other calculators.\r
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\n" ); document.write( "\n" ); document.write( "Then,
\n" ); document.write( "f^1 = integral[ f^2 ]
\n" ); document.write( "f^1 = integral[ -5t^2 - (20/3)t^(3/2) + C ]
\n" ); document.write( "f^1 = (-5/3)t^3 - (20/3)*(1/(3/2+1))*t^(3/2+1) + C*t + D
\n" ); document.write( "f^1 = (-5/3)t^3 - (8/3)*t^(5/2) + C*t + D\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "As a check,
\n" ); document.write( "f^2 = derivative[ f^1 ]
\n" ); document.write( "f^2 = derivative[ (-5/3)t^3 - (8/3)*t^(5/2) + C*t + D ]
\n" ); document.write( "f^2 = (-5/3)*3*t^(3-1) - (8/3)*(5/2)*t^(5/2-1) + C
\n" ); document.write( "f^2 = -5*t^2 - (20/3)*t^(3/2) + C\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Lastly,
\n" ); document.write( "f = integral[ f^1 ]
\n" ); document.write( "f = integral[ (-5/3)t^3 - (8/3)*t^(5/2) + C*t + D ]
\n" ); document.write( "f = (-5/3)*(1/(3+1))*t^(3+1) - (8/3)*(1/(5/2+1))*t^(5/2+1) + (C/2)*t^2 + D*t + E
\n" ); document.write( "f = (-5/12)*t^4 - (16/21)*t^(7/2) + (C/2)*t^2 + D*t + E\r
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\n" ); document.write( "\n" ); document.write( "I'll leave the derivative check for the student to carry out.\r
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\n" ); document.write( "\n" ); document.write( "Summary
  • original function = f = (-5/12)*t^4 - (16/21)*t^(7/2) + (C/2)*t^2 + D*t + E
  • 1st derivative = f^1 = (-5/3)t^3 - (8/3)*t^(5/2) + C*t + D
  • 2nd derivative = f^2 = -5*t^2 - (20/3)*t^(3/2) + C
where C, D, and E are constants. Each answer has been verified.
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