5 Steps to a 5 AP Calculus BC 2019

More Applications of Derivatives 179

Example 5

Using your calculator, find the value(s) ofxto the nearest hundredth at which the slope

of the line tangent to the graph ofy=2ln(x^2 +3) is equal to−

1

2

. (See Figures 9.1-8 and
9.1-9.)

[−5, 5] by [−1, 7]

Figure 9.1-8

[−10, 3] by [−1, 10]

Figure 9.1-9

Step 1: Entery 1 = 2 ∗ln (x∧ 2 +3).

Step 2: Entery 2 =d(y 1 (x),x) and entery 3 =−

1

2

.

Step 3: Using the [Intersection] function of the calculator fory 2 andy 3 , you obtainx=

− 7 .61 orx=− 0 .39.

Example 6

Using your calculator, find the value(s) ofxat which the graphs ofy= 2 x^2 andy=exhave

parallel tangents.

Step 1: Find

dy

dx

for bothy= 2 x^2 andy=ex.

y= 2 x^2 ;

dy

dx

= 4 x

y=ex;

dy

dx

=ex

Step 2: Find thex-coordinate of the points of tangency. Parallel tangents⇒slopes are

equal.

Set 4x=ex⇒ 4 x−ex=0.

Using the [Solve] function of the calculator, enter [Solve](4x−ê(x)=0, x) and

obtainx= 2 .15 andx= 0 .36.

TIP • Watch out for different units of measure, e.g., the radius,r, is 2 feet, finddr
dt

in inches

per second.