5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:14

204 STEP 4. Review the Knowledge You Need to Score High

[–.5π, π] by [–4, 4]

Figure 10.1-4

y=−3 sin 2x;

dy

dx

=−3[cos(2x)]2=−6 cos(2x)

Slope of tangent

(

atx=

π

2

)

:

dy

dx

∣∣

∣∣

x=π/ 2

=−6 cos [2(π/2)]=−6 cosπ= 6.

Point of tangency: Atx=

π

2

,y=−3 sin(2x)

=−3 sin[2(π/2)]=−3 sin(π)= 0.

Therefore,

(

π

2

,0

)

is the point of tangency.

Equation of tangent:y− 0 =6(x−π/2) ory= 6 x− 3 π.

Example 2

If the liney= 6 x+ais tangent to the graph ofy= 2 x^3 , find the value(s) ofa.

Solution:

y= 2 x^3 ;

dy

dx

= 6 x^2 .(See Figure 10.1-5.)

[–2, 2] by [–6, 6]

Figure 10.1-5

The slope of the liney= 6 x+ais 6.

Sincey= 6 x+ais tangent to the graph ofy= 2 x^3 , thus

dy

dx

=6 for some values ofx.