Question

Find the values of a and b if the tangent line to f(t)=t^3+at+b at t=3 is y=5t-7.

Answer 1

\[f \left( t \right)=t ^{3}+at+b\]
\[f \prime \left( t \right)=3t ^{2}+a\]
\[at t=3,f \prime \left( t \right)=3\left( 3 \right)^{2}+a=27+a\]
slope of tangent at t=3 is 27+a
Also eq. of tangent is y=5t-7
\[\frac{ dy }{dt}=5,at t=3,slope of line =5\]
27+a=5
a=-22
at t=3,y=5*3-7=8
\[Hence 8=3^{3}-22*3+b\]
8=27-66+b
b=8+66-27=74-27=47

Answer 2

Thank you so much! I was lost on how to begin this and I can see I even attempted the problem wrong.