L 60 1. y=x^2 +5x Find the equation of a line with the slope of 7 that touches y=x^2 +5x. [Sol.] Let f(x)=x^2 +5x Then f′(x)=2x+5 Since 2x+5=7⇒x=1 then the point is (1, 1^2 +5∙1)=(1, 6) So the equation of a line is y−6=7(x−1)⇒y=7x−1 2. y=ax^2 +bx (2, 2) a, b Find the values of constants a, b that the slope of the line that touches (2, 2) and y=ax^2 +bx is 5. [Sol.] Let f(x)=ax^2 +bx Then f′(x)=2ax+b and build two equations to solve for a and b { ((f(2)=a∙2^2 +b∙2=4a+2b=2)),((f′(2)=2a∙2+b=4a+b=5)) :} Solving for a, b gives a=2, b=−3 3. y=x^3 −3x^2 −1 Find the equation of a line that is drawn, touches y=x^3 −3x^2 −1. [Sol.] The line of the equation is y−(a^3 −3a^2 −1)=(3a^2 −6a)(x−a) Calculating gives y=(3a^2 −6a)x−(3a^2 −6a)a+(a^3 −3a^2 −1) y=(3a^2 −6a)x+(−3a^3 +6a^2 )+(a^3 −3a^2 −1) y=(3a^2 −6a)x+(−2a^3 +3a^2 −1) −2a^3 +3a^2 −1=0 a=−(1/2) or a=2 ...a=−3x, a=((15)/4)x