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(16) 1. Calculate the following limits. Give a brief justiî€Ścation of your answers without

reference to calculator computations or graphing.

(a) lim

x!0

4x

sin3x

lim

x!0

4x

sin3x

= lim

x!0

4

3

sin 3x

3x

=

4

3

1

=

4

3

(b) lim

x!î€€1

3x

4

+x

3

î€€2x

2

+10

7x

5

î€€4x

3

+2x+1

lim

x!î€€1

3x

4

+x

3

î€€2x

2

+10

7x

5

î€€4x

3

+2x+1

= lim

x!î€€1

3

x

+

1

x

2

î€€

2

x

3

+

10

x

5

7î€€

4

x

2

+

2

x

4

+

1

x

5

=

0

7

=0

(c) lim

x!0

cos(2x)î€€1

x

2

Using l'H^opital's Rule twice,

lim

x!0

cos(2x)î€€1

x

2

= lim

x!0

î€€2sin(2x)

2x

= lim

x!0

î€€4cos(2x)

2

=

î€€4

2

=î€€2

(d) lim

x!î€€3

î€€

j2x+6j

x+3

If xis slightly less than î€€3, then 2x+3<0, so j2x+6j=î€€2xî€€6. Thus

lim

x!î€€3

î€€

j2x+6j

x+3

= lim

x!î€€3

î€€

î€€2(x+3)

x+3

= lim

x!î€€3

î€€

î€€2=î€€2

(10) 2. Compute the derivative of

2

xî€€1

directly from the deî€Śnition.

î€’

2

xî€€1

î€“

0

= lim

h!0

2

x+hî€€1

î€€

2

xî€€1

h

= lim

h!0

2(xî€€1) î€€2(x+hî€€1)

h(x+hî€€1)(xî€€1)

=

lim

h!0

î€€2h

h(x+hî€€1)(xî€€1)

= lim

h!0

î€€2

(x+hî€€1)(xî€€1)

=

î€€2

(xî€€1)

2

bvnbvnbv

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2A

(16) 3. Compute the derivatives with respect to xof the following functions. Algebraic

simpliî€Ścation of the answers need not be performed.

(a) e

x

ln(2x)

e

x

î€’

1

2x

î€“

2+e

x

ln(2x)

(b)

sinx

x

3

+2x

(x

3

+2x)cos(x)î€€sin(x)(3x

2

+2)

(x

3

+2x)

2

(c)

Z

0

x

sectdt

î€’

Z

0

x

sectdt

î€“

0

=

î€’

î€€

Z

x

0

sectdt

î€“

0

=î€€secx

(d)

Z

x

2

+x

0

sin(2t)dt

sin(2(x

2

+x))(2x+1)

(10) 4. Suppose that fis a function with î€Śrst and second derivatives. Suppose in addition

that the following values are known: f(1) = 0, f

0

(1) = 3, and f

00

(1) = 5. If g(x)=e

f(x)

,

what are g

0

(1) and g

00

(1)?

g

0

(x)=e

f(x)

f

0

(x)

so g

0

(1) = e

0

(3) = 3.

g

00

(x)=e

f(x)

f

00

(x)+e

f(x)

f

0

(x)

2

Thus g

00

(1) = e

0

(5) + e

0

(3)

2

=14

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3A

(15) 5. Find the following indeî€Śnite integrals:

(a)

Z

î€’

x

4

+ sec(x)tan(x)î€€

2

x

î€“

dx

x

5

5

+ secxî€€2lnjxj+C

(b)

Z

3x

2

sin(x

3

+1)dx

If u=x

3

+1,then du =3x

2

dx and

Z

3x

2

sin(x

3

+1)dx =

Z

sin(u)du =î€€cos(u)+C=î€€cos(x

3

+1)+C

(c)

Z

cos(x)e

sin(x)

dx

If u= sin(x), then du = cos(x)dx and

Z

cos(x)e

sin(x)

dx =

Z

e

u

du =e

u

+C=e

sin(x)

+C

(18) 6. Compute the following:

(a)

Z

2

1

x

3

+5

x

dx

Z

2

1

x

3

+5

x

dx =

Z

2

1

î€’

x

2

+

5

x

î€“

dx =

î€’

x

3

3

+5lnx

î€“

2

1

=

8

3

+ 5ln2 î€€

î€’

1

3

+ 5ln1

î€“

=

7

3

+ 5ln2

(b) The area under the graph of y=4x+e

x

on the interval [0;2].

The area is

Z

2

0

(4x+e

x

)dx =(2x

2

+e

x

)j

2

0

=8+e

2

î€€1=7+e

2

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