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

MA 3972-MA-Book April 11, 2018 15:57

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

Letu=x^2 ; then

du

dx

= 2 x.

Rewrite:y=−

∫u

1

sintdt.

dy

dx

=

dy

du

·

du

dx

=(−sinu)2x=(−sinx^2 )2x

=− 2 xsin(x^2 )

Example 5

Find

dy

dx

;ify=

∫x 2

x

√

et+ 1 dt.

y=

∫ 0

x

√

et+ 1 dt+

∫x 2

0

√

et+ 1 dt=−

∫x

0

√

et+ 1 dt+

∫x 2

0

√

et+ 1 dt

=

∫x 2

0

√

et+ 1 dt −

∫x

0

√

et+ 1 dt

Sincey=

∫ x 2

0

√

et+ 1 dt−

∫ x

0

√

et+ 1 dt

dy

dx

=

(

d

dx

∫x 2

0

√

et+ 1 dt

)

−

(

d

dx

∫x

0

√

et+ 1 dt

)

=

(√

ex^2 + 1

)

d

dx

(x^2 )−

(√

ex+ 1

)

= 2 x

√

ex^2 + 1 −

√

ex+ 1.

Example 6

F(x)=

∫x

1

(t^2 −4)dt, integrate to findF(x) and then differentiate to findf′(x).

F(x)=

t^3

3

− 4 t

]x

1 =

(

x^3

3

− 4 x

)

−

(

13

3

−4(1)

)

=

x^3

3

− 4 x+

11

3

F′(x)= 3

(

x^2

3

)

− 4 =x^2 − 4