2. Let functions p and q be the piecewise linear functions given by their respec

Question

2. Let functions p and q be the piecewise linear functions given by their respective graphs in Figure 2.1. Use the graphs to answer the following questions. -3 -1 -2 Figure 2.1: The graphs of p (in blue) and q (in green). (a) At what values of x is p not differentiable? At what values of x is q not differentiable? Why? (b) Let r(x) = p(x) + 2g(x). At what values of x is r not differentiable? Why? (c) Determine r'(-2) and r'(0) (d) Find an equation for the tangent line to y = r(x) at the point (2,r(2)).

Explanation / Answer

At points x=-1 andd x=1 p is not differentiable as slope of lines different at both sides.

At points x=-1 andd x=1 q is not differentiable as slope of lines different at both sides.

We need to check differentiablity of r at -1 and 1 as all remaining points it would be diffferentiable.

r'(x) = p'(x) + 2q'(x)

R'(just before -1) = -2 + 2 *3 = 4

R'(just after -1) = 0.5 + 0 = 0.5

R'(just before 1) = 0.5 + 0 = 0.5

R'(just after 1) = 2 + 2*(-1) = 0

Hence at 1,-1 it is not differentiable as slope of lines different at both sides.

r'(-2) = p'(-2) + 2q'(-2) = -2 + 2*3 = 4

r'(0) = p'(0) + 2q'(0) = 0.5 + 0 = 0.5

r'(2) = p'(2) + 2q'(2) = 2 + 2*(-1) = 0

r(2) = p(2) + 2q(2) = 2+1*2 = 4

Line ==> (y-4) = 0(x-2) ====> y=4

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