Continuity and Differentiability — Course Quiz

The greatest integer function

[x]

is discontinuous at

A

x = 1.7

B

x = 2

C

x = 0.5

D

x = 2.5

Find

a

so that

f(x) = \begin{cases} ax + 1, & x \le 2 \\ 3x - 1, & x > 2 \end{cases}

is continuous.

A

3

B

\displaystyle \tfrac{3}{2}

C

\displaystyle \tfrac{5}{2}

D

2

Which statement is true?

A Every continuous function is differentiable.

B

\lvert x \rvert

is differentiable at

0

.

C

\lvert x \rvert

is discontinuous at

0

.

D Every differentiable function is continuous.

\displaystyle \dfrac{d}{dx}(3x - 4)^5

equals

A

15(3x - 4)^4

B

15(3x - 4)^5

C

3(3x - 4)^4

D

5(3x - 4)^4

\displaystyle \dfrac{d}{dx}e^{x^3}

equals

A

3x^2e^{x^3}

B

e^{x^3}

C

3xe^{x^3}

D

x^3e^{x^3 - 1}

\displaystyle \dfrac{d}{dx}\sin^{-1}(3x)

equals

A

\displaystyle \dfrac{3}{\sqrt{1 - 3x^2}}

B

\displaystyle \dfrac{1}{\sqrt{1 - 9x^2}}

C

\displaystyle \dfrac{3}{\sqrt{1 - 9x^2}}

D

\displaystyle \dfrac{3}{1 + 9x^2}

\displaystyle \dfrac{d}{dx}5^x

equals

A

\displaystyle \dfrac{5^x}{\log 5}

B

5^x\log 5

C

5^x

D

x \cdot 5^{x - 1}

If

y = x^x

, find

\displaystyle \dfrac{dy}{dx}

at

x = 1

.

A

1

B

0

C

2

D

e

If

y = A\sin 3x + B\cos 3x

, which equation does

y

satisfy?

A

y' + 3y = 0

B

y'' + 9y = 0

C

y'' + 3y = 0

D

y'' - 9y = 0

If

x^2 + y^2 = 16

, then

\displaystyle \dfrac{dy}{dx}

equals

A

\displaystyle -\dfrac{x}{y}

B

\displaystyle \dfrac{x}{y}

C

\displaystyle -\dfrac{y}{x}

D

2x + 2y

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Continuity and Differentiability — Course Quiz

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