How would you solve to find the equation of the tangent line to the curve #for y = (1+x) cos x# at the given point (0,1)?

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Answer 1 · 2018-06-30T03:58:19

#y=x+1#

#y=(1+x)cosx#

First let's verify that #(0,1)# is a point on #y#

#y(0)= (1+0)cos(0) = 1xx1=1 -> y(0) =1#

#dy/dx= (1+x)*(-sinx) + 1*cosx#

#= cosx-(1+x)sinx#

#dy/dx# at #x=0# will give us the slope of #y# at #(0,1)#

Call this slope #m#

#-> m = cos(0) - (1-0)sin(0) = 1-1xx0 =1#

Now, the tangent to #y# at #(0,1)# will have the equation:

#y=mx+c# where #m=1# (from above)

Since, #(0,1)# is a point on this line

#:.1=1xx0+c -> c=1#

Hence, our tangent has the equation: #y=x+1#

We can see the graph of #y# (Red) and our tangent (Blue) in the graphic below.

enter image source here