Final Exam Revision 24/25 t1

Match

Match the following

Indicate whether the following statements are true or false. Justify your answer.

False = 4

False= 2

True

• False = 4

• False= 2

• True

Multiple Choice

Evaluate $$\lim_{x\rightarrow\infty}\ \frac{3x^2-5x+8}{2x-7}$$  

• $$0$$  

• $$\infty$$  

• $$\frac{3}{2}$$  

• $$-\infty$$  

• $$\frac{5}{7}$$  

Multiple Choice

Evaluate $$\lim_{x\rightarrow\infty}\ \frac{2x-3}{4x\ +1}$$  

• $$0$$  

• $$Does\ not\ exist$$  

• $$\frac{1}{2}$$  

• $$2$$  

• $$4$$  

Multiple Choice

Evaluate $$\lim_{x\rightarrow\infty}\ \frac{4x-3}{2x\ +1}$$  

• $$0$$  

• $$Does\ not\ exist$$  

• $$\frac{1}{2}$$  

• $$2$$  

• $$4$$  

Multiple Choice

Evaluate $$\lim_{x\to\infty}\frac{3x-2}{6-x}=$$  

• $$\frac{1}{2}$$  

• $$-\frac{1}{3}$$  

• $$-3$$  

• $$\frac{1}{3}$$  

• $$3$$  

Multiple Choice

Evaluate $$\lim_{x\to\infty}\frac{3x^2-2}{6x^3-x}=$$  

• $$\infty$$   

• $$\frac{1}{2}$$  

• $$0$$  

• $$-\frac{1}{3}$$  

• $$-\infty$$  

Multiple Choice

Evaluate $$\lim_{x\to-\infty}\frac{3x^2-2x^5}{6x^3-x^4}=$$  

• $$\infty$$  

• $$2$$  

• $$\frac{1}{2}$$  

• $$-\frac{1}{3}$$  

• $$-\infty$$  

Multiple Choice

Evaluate $$\lim_{x\rightarrow\infty}\left(2x^3+3x-1\right)$$  

• $$\infty$$  

• $$-\infty$$  

• $$2$$  

• $$\frac{2}{3}$$  

• $$3$$  

Multiple Choice

Evaluate $$\lim_{x\rightarrow\infty}\left(-2x^3+3x-1\right)$$  

• $$\infty$$  

• $$-\infty$$  

• $$2$$  

• $$\frac{2}{3}$$  

• $$3$$  

Multiple Choice

Evaluate $$\lim_{x\rightarrow-\infty}\left(2x^3+3x-1\right)$$  

• $$\infty$$  

• $$-\infty$$  

• $$2$$  

• $$\frac{2}{3}$$  

• $$3$$  

Multiple Choice

Evaluate $$\lim_{x\rightarrow-\infty}\left(-2x^3+3x-1\right)$$  

• $$\infty$$  

• $$-\infty$$  

• $$2$$  

• $$\frac{2}{3}$$  

• $$3$$  

Multiple Choice

Find the vertical asymptotes (if any) of the graph of the function.

$$f\left(x\right)=\frac{2}{\left(x-3\right)^3}$$   

• $$x=0$$  

• $$x=0\ and\ x=2$$  

• $$x=3$$   

• $$x=-2$$  

• $$x=0\ and\ x=-2$$  

Multiple Choice

Find the vertical asymptotes (if any) of the graph of the function.

$$f\left(x\right)=\frac{x^2}{x^2-4}$$    

• $$x=0$$  

• $$x=-2\ and\ x=2$$   

• $$x=3$$   

• $$x=-2$$  

• $$x=0\ and\ x=-2$$  

Multiple Choice

Find the vertical asymptotes (if any) of the graph of the function.

$$f\left(x\right)=\frac{3}{x^2+x-2}$$     

• $$x=-2\ and\ x=1$$   

• $$x=-2\ and\ x=2$$   

• $$x=3$$   

• $$x=-2$$  

• $$x=0\ and\ x=-2$$  

Multiple Choice

Evaluate : $$\lim_{x\rightarrow\infty}\left(3x^5+5x^2-3x\right)$$  

• $$\infty$$  

• $$-\infty$$

• $$0$$  

• $$3$$  

• $$DNE$$  

Multiple Choice

Evaluate : $$\lim_{x\rightarrow\infty}\left(2x-4x^6\right)$$   

• $$\infty$$  

• $$-\infty$$

• $$0$$  

• $$3$$  

• $$DNE$$  

Multiple Choice

Evaluate : $$\lim_{x\rightarrow-\infty}\left(-5x^7-3x^3+11\right)$$   

• $$\infty$$  

• $$-\infty$$

• $$0$$  

• $$3$$  

• $$DNE$$  

Multiple Choice

Evaluate : $$\lim_{x\rightarrow-\infty}\left(\frac{x^2-3x}{4-x^2}\right)$$   

• $$\infty$$  

• $$-\infty$$

• $$0$$  

• $$-1$$   

• $$DNE$$  

Multiple Choice

Use numerical evidence to conjecture value for the limit.

$$\lim_{x\rightarrow0}\ \frac{3x^2+9x}{\sin\left(x\right)}$$  

• $$8$$  

• $$8.7$$  

• $$9.3$$  

• $$9$$  

• $$DNE$$  

Multiple Choice

Use numerical evidence to conjecture value for the limit.

$$\lim_{x\rightarrow0}\ \frac{4x^2+8x}{\sin\left(x\right)}$$  

• $$8$$  

• $$9.3$$  

• $$8.7$$  

• $$9$$  

• $$DNE$$  

Match

Match the following

Use the graph of f(x) to answer the following questions.

• $$f\left(-2\right)$$  

• $$f\left(-1\right)$$  

• $$\lim_{x\rightarrow-2}\ f\left(x\right)$$  

• $$\lim_{x\rightarrow-1}\ f\left(x\right)$$

3

UNDEFINED

2

0

• 3

• UNDEFINED

• 2

• 0

Multiple Choice

Use the graph of f(x) to answer the following question.

$$\lim_{x\rightarrow-2}\ f\left(x\right)$$  

• $$2$$  

• $$4$$  

• $$3$$  

• $$DNE$$  

• $$0$$  

Multiple Choice

Use the graph of f(x) to answer the following question.

$$\lim_{x\rightarrow-2^-}\ f\left(x\right)$$  

• $$2$$  

• $$4$$  

• $$3$$  

• $$DNE$$  

• $$0$$  

Multiple Choice

Use the graph of f(x) to answer the following question.

$$\lim_{x\rightarrow-2^+}\ f\left(x\right)$$  

• $$2$$  

• $$4$$  

• $$3$$  

• $$DNE$$  

• $$0$$  

Multiple Choice

Use the graph of f(x) to answer the following question.

$$\lim_{x\rightarrow0^+}\ f\left(x\right)$$  

• $$2$$  

• $$4$$  

• $$3$$  

• $$DNE$$  

• $$0$$  

Match

Match the following

Use the graph of f(x) to answer the following questions.

• $$f\left(0\right)$$  

• $$f\left(2\right)$$  

• $$\lim_{x\rightarrow-2}\ f\left(x\right)$$  

• $$\lim_{x\rightarrow-1}\ f\left(x\right)$$

2

UNDEFINED

2

0

• 2

• UNDEFINED

• 2

• 0

Match

Match the following 

• $$f\left(2\right)$$  

• $$\lim_{x\rightarrow2^-}f\left(x\right)$$  

• $$\lim_{x\rightarrow2^+}f\left(x\right)$$

• $$\lim_{x\rightarrow2^{ }}f\left(x\right)$$

• $$f\left(3\right)$$  

Undefined

-∞

∞

DNE

1

• Undefined

• -∞

• ∞

• DNE

• 1

Multiple Choice

Use a graph to estimate  $$\lim_{x\rightarrow-3}\ \frac{-1}{\left(x+3\right)^2}$$   if it exists.               

• $$-3$$  

• $$\infty$$  

• $$-\infty$$  

• $$0$$  

• $$-8$$   

Multiple Choice

Use $$\lim_{x\rightarrow a}\ f\left(x\right)=8\ and\ \lim_{x\rightarrow a}\ g\left(x\right)=-8$$  

to determine the limit.

$$\lim_{x\rightarrow a}\ \left[2f\left(x\right)g\left(x\right)\right]$$  

• $$-128$$  

• $$128$$  

• $$24$$  

• $$8$$  

• $$-2$$  

Multiple Choice

Use $$\lim_{x\rightarrow a}\ f\left(x\right)=8\ and\ \lim_{x\rightarrow a}\ g\left(x\right)=-8$$  

to determine the limit.

$$\lim_{x\rightarrow a}\ \left[-2f\left(x\right)g\left(x\right)\right]$$  

• $$-128$$  

• $$128$$  

• $$24$$  

• $$8$$  

• $$-2$$  

Multiple Choice

Use $$\lim_{x\rightarrow a}\ f\left(x\right)=8\ and\ \lim_{x\rightarrow a}\ g\left(x\right)=-8$$  

to determine the limit.

$$\lim_{x\rightarrow a}\ \left[-2f\left(x\right)-g\left(x\right)\right]$$  

• $$-128$$  

• $$128$$  

• $$24$$  

• $$-8$$  

• $$-2$$  

Multiple Choice

$$Evaluate\ :\ \lim_{x\rightarrow3}\ \ \frac{x^2+2x-15}{x^2-x-6}\ =\ ....$$  

• $$\frac{5}{8}$$  

• $$\frac{8}{5}$$  

• $$-\frac{8}{5}$$  

• $$-\frac{5}{8}$$  

• 0

Multiple Choice

$$Evaluate\ :\lim_{x\rightarrow2}\left(2x^3+5x^2+2x+1\right)=....$$  

• $$38$$  

• $$39$$  

• $$50$$  

• $$41$$  

• $$76$$  

Multiple Choice

$$Evaluate\ :\lim_{x\rightarrow1}\ \frac{x^2\ -\ 5x\ +\ 4}{x\ -\ 1}\$$  

• $$-\ 5$$

• $$-\ 4$$

• $$-\ 3$$

• $$0$$

• $$5$$

Multiple Choice

$$Evaluate\ :\lim_{x\rightarrow2}\ \frac{x^2-4}{x-2}=.....$$  

• $$0$$  

• $$4$$  

• $$DNE$$  

• $$-2$$  

• $$-4$$  

Multiple Choice

$$Evaluate\ :\lim_{x\rightarrow2}\ \ \frac{x^2-5x+6}{x^2+2x-8}\ =\ ....$$  

• 2

• 1

• $$\frac{1}{6}$$  

• $$-\frac{1}{6}$$  

• $$-\frac{1}{12}$$  

Multiple Choice

$$Evaluate\ :\lim_{x\rightarrow3}\left(2x-5\right)=....$$  

• 3

• 1

• -1

• -3

• 7

Multiple Choice

Evaluate: $$\lim_{x\rightarrow1}\left(\frac{x-2}{x+3}\right)=$$  

• $$-\frac{2}{3}$$  

• $$-\frac{1}{4}$$  

• $$\frac{1}{2}$$  

• $$\frac{2}{3}$$  

• 1

Multiple Choice

$$\lim_{x\rightarrow-2}\left(\frac{x+2}{x^2+5x+6}\right)$$  

• ∞
• 0

• 3
• 1

Multiple Choice

$$\lim_{x\rightarrow-4}\ \ \ 17\ =\ ....$$  

• 17

• $$-68$$  

• $$-4$$  

• $$-17$$  

• 0

Multiple Choice

Evaluate the indicated limit, if it exists.

• $$\lim_{x\rightarrow8}\ f\left(x\right)=0$$  

• $$\lim_{x\rightarrow8}\ f\left(x\right)=2$$

• $$\lim_{x\rightarrow8}\ f\left(x\right)does\ not\ exists$$

• $$\lim_{x\rightarrow8}\ f\left(x\right)=64$$

• $$\lim_{x\rightarrow8}\ f\left(x\right)=8$$

Multiple Choice

Evaluate the indicated limit, if it exists.

• $$\lim_{x\rightarrow2}\ f\left(x\right)=0$$  

• $$\lim_{x\rightarrow2}\ f\left(x\right)=2$$

• $$\lim_{x\rightarrow2}\ f\left(x\right)does\ not\ exists$$

• $$\lim_{x\rightarrow2}\ f\left(x\right)=64$$

• $$\lim_{x\rightarrow2}\ f\left(x\right)=4$$

Multiple Choice

Evaluate the indicated limit, if it exists.

• $$\lim_{x\rightarrow6}\ f\left(x\right)=0$$  

• $$\lim_{x\rightarrow6}\ f\left(x\right)=2$$

• $$\lim_{x\rightarrow6}\ f\left(x\right)does\ not\ exists$$

• $$\lim_{x\rightarrow6}\ f\left(x\right)=64$$

• $$\lim_{x\rightarrow6}\ f\left(x\right)=36$$

Multiple Choice

$$\lim_{x\rightarrow-4}\ \ \ -17\ =\ ....$$  

• $$17$$  

• $$-68$$  

• $$-4$$  

• $$-17$$  

• 0

Multiple Choice

$$\lim_{x\rightarrow-4}\ \ \ 0\ =\ ....$$  

• $$17$$  

• $$-68$$  

• $$-4$$  

• $$-17$$  

• $$0$$  

Multiple Choice

$$Evaluate\ :\lim_{x\rightarrow2}\left(2x^3+5x^2+2x-2\right)=....$$  

• $$38$$  

• $$39$$  

• $$50$$  

• $$41$$  

• $$76$$  

Multiple Choice

Use the graph of f(x) to answer the following question.

$$\lim_{x\rightarrow0}f\left(x\right)$$  

• $$4$$  

• $$-1$$  

• $$0$$  

• $$DNE$$  

• 3

Multiple Choice

Use the graph of f(x) to answer the following question.

$$\lim_{x\rightarrow-2}f\left(x\right)$$  

• $$4$$  

• $$-1$$  

• $$0$$  

• $$DNE$$  

• $$8$$  

Multiple Choice

Let f(x) be the function whose graph is shown below.

Which of the following statements best describes why f(x) is discontinuous at x = c?

• $$\lim_{x\rightarrow c}f\left(x\right)\ne f\left(c\right)$$  

• $$\lim_{x\rightarrow c}f\left(x\right)\ does\ not\ exost$$

• $$f\left(c\right)$$ undefined

• $$\lim_{x\rightarrow c^-}f\left(x\right)\ne\lim_{x\rightarrow c^+}f\left(x\right)$$

• Infinite discontinuity

Multiple Choice

Evaluate

$$f\left(2\right)$$  

• 2

• 1

• 4

• 5

Multiple Choice

Evaluate

$$\lim_{x\rightarrow2^-}f\left(x\right)$$  

• 1

• 2

• 3

• DNE

Multiple Choice

Evaluate

$$\lim_{x\rightarrow2^+}f\left(x\right)$$  

• 1

• 2

• 3

• DNE

Multiple Choice

Evaluate

$$\lim_{x\rightarrow2^{ }}f\left(x\right)$$  

• 1

• 2

• 3

• DNE

Match

Match the following

• $$f\left(2\right)$$  

• $$\lim_{x\rightarrow2^-}f\left(x\right)$$  

• $$\lim_{x\rightarrow2^+}f\left(x\right)$$

• $$\lim_{x\rightarrow2^{ }}f\left(x\right)$$

Undefined

1

3

DNE

• Undefined

• 1

• 3

• DNE

Match

Match the following 

• $$f\left(2\right)$$  

• $$\lim_{x\rightarrow2^-}f\left(x\right)$$  

• $$\lim_{x\rightarrow2^+}f\left(x\right)$$

• $$\lim_{x\rightarrow2^{ }}f\left(x\right)$$

Undefined

-∞

∞

DNE

• Undefined

• -∞

• ∞

• DNE

Multiple Choice

Which one is the product rule?

• $$f\left(x\right)g\left(x\right)$$  

• $$f'(x)g(x)-f(x)g'(x)$$  

• $$f'(x)g(x)+f(x)g'(x)$$  

• $$f'(x)g'(x)+f(x)g(x)$$  

Multiple Choice

$$\frac{\text{d}}{\text{d}x}\left(x^2\right)=?$$  

• $$2x$$  

• $$x^2$$  

• 1

• 0

Fill in the Blank

$$\frac{\text{d}}{\text{d}x}\left(x^3\right)=?$$  

^

Multiple Choice

$$\frac{d}{dx}\left(x^n\right)$$  

• $$x^{n-1}$$  

• $$n\ x^{n-1}$$  

• $$n\ x$$  

• $$x^n$$  

Multiple Choice

Find the derivative of the given equation

f(x) = x⁴ + x³ + 100

• 4x³ + 3x² + 100

• 4x³ + 3x²

• 4x⁵ + 3x⁴ + 100x

• x³ + x² + 100

Multiple Choice

Find the difference quotient for   $$f\left(x\right)=3x$$  

• $$3h$$  

• $$3x$$  

• $$3$$  

• $$3\left(x+h\right)$$  

Multiple Choice

Find the derivative of $$f\left(x\right)\ =\ \frac{1}{2\sqrt{x}}$$  

• $$-\frac{1}{4\sqrt{x^3}}$$  

• $$-\frac{1}{\sqrt{x^3}}$$  

• $$-x^2$$  

• $$-4x^2$$  

Multiple Choice

Find the derivative of the given function:
$$f\left(x\right)=\frac{1}{x^2}$$

• $$f'\left(x\right)=\frac{1}{2x}$$  

• $$f'\left(x\right)=-\frac{2}{x^3}$$  

• $$f'\left(x\right)=-\frac{2}{x}$$  

• $$f'\left(x\right)=-\frac{1}{2x^3}$$  

Multiple Choice

Find the derivative of $$y=\frac{12}{x^2}$$  

• ^{$$y'=\frac{24}{x}$$}  

• ^{$$y'=-\frac{24}{x^3}$$}  

• ^{$$y'=-\frac{24}{x}$$}  

• ^{$$y'=\frac{6}{x^3}$$}  

Multiple Choice

Find the derivative of the function: $$f\left(x\right)=\sqrt{x^5}$$  

• $$f'\left(x\right)=\frac{5}{2\sqrt{x^3}}$$  

• $$f'\left(x\right)=\frac{5\sqrt{x^3}}{2}$$  

• $$f'\left(x\right)=-\frac{5}{2\sqrt{x^3}}$$  

• $$f'\left(x\right)=-\frac{5\sqrt{x^3}}{2}$$  

Multiple Choice

Find the derivative of the function: $$f\left(x\right)=\frac{1}{\sqrt{x^3}}$$

• $$f'\left(x\right)=-\frac{3}{2\sqrt{x^5}}$$  

• $$f'\left(x\right)=-\frac{2}{3\sqrt{x^5}}$$  

• $$f'\left(x\right)=-\frac{3}{\sqrt{x}}$$  

• $$f'\left(x\right)=-\frac{3}{2\sqrt{x}}$$  

Multiple Choice

Differentiate: $$y=\sqrt[3]{x^4}$$  

• $$\frac{dy}{dx}=\frac{4}{3}\sqrt[3]{x}$$  

• $$\frac{dy}{dx}=\frac{3}{4\sqrt[4]{x}}$$  

• $$\frac{dy}{dx}=\frac{4}{3\sqrt[3]{x}}$$  

• $$\frac{dy}{dx}=\frac{3}{4}\sqrt[4]{x}$$  

Multiple Choice

$$f\left(x\right)=\frac{7}{8}x^4+2x$$  

• $$f'\left(x\right)=\frac{7}{2}x^2+2$$  

• $$f'\left(x\right)=\frac{7}{2}x^3+2x$$  

• $$f\left(x\right)=\frac{7}{2}x^3+3$$  

• $$f'\left(x\right)=\frac{28}{8}x^3+2x$$  

Multiple Choice

$$f\left(x\right)=-3x^4-6x+8$$  . Find $$f'\left(x\right)$$  .

• $$12x^3$$  

• $$-12x^3-6$$  

• $$12x^3+6+8$$  

• $$-12x^5-6x^2+8x$$  

Multiple Choice

$$f\left(x\right)=x^{-6}$$  . Find $$f'\left(x\right)$$  and write it in simplified form.

• $$-6x^{-7}$$  

• $$\frac{-6}{x^7}$$  

• $$-6x^{-5}$$  

• $$\frac{-6}{x^{-7}}$$  

Multiple Choice

$$f\left(x\right)=x^{-1}$$  . Find f'(x) and write it in simplified form.

• $$-x^2$$  

• $$-x^{-2}$$  

• $$\frac{-1}{x^2}$$  

• $$\frac{-2}{x^2}$$  

Multiple Choice

$$f\left(x\right)=\sqrt[]{x}$$  Find f'(x) and write it in the most simplified version.

• $$\frac{1}{2\ \sqrt[]{x}}$$   

• $$\frac{1}{2}x^{-\frac{1}{2}}$$  

• $$\frac{1}{2x^{\frac{1}{2}}}$$  

• This is impossible to know, stop torturing me with derivatives.

Multiple Select

$$f\left(x\right)=4\ \sqrt[3]{x}$$  . This time select ALL of the versions that are correct, not just the most official one.

• $$\frac{4}{3}x^{-\frac{2}{3}}$$  

• $$\frac{4}{3\ x^{\frac{2}{3}}}$$  

• $$\frac{4}{3\ \ \sqrt[3]{x^2}}$$  

• $$\frac{4}{3\ \left(\sqrt[3]{x}\right)^2}$$  

Multiple Choice

$$f\left(x\right)=5\sqrt[]{x}$$  . You know the deal. Simplified version.

• $$\frac{5}{2\ \sqrt[]{x}}$$  

• $$\frac{5}{2}x^{-\frac{1}{2}}$$  

• $$\frac{10}{\sqrt[]{x}}$$  

• $$\frac{5}{2}\left(-\sqrt[]{x}\right)$$  

Multiple Select

$$f\left(x\right)=x^{\frac{5}{7}}$$  . Pick all that are valid.

• $$\frac{5}{7\ x^{\frac{2}{7}}}$$  

• $$\frac{5}{7}x^{-\frac{2}{7}}$$  

• $$\frac{5}{7\ \sqrt[7]{x^2}}$$  

• $$\frac{5}{7\ \sqrt[2]{x^7}}$$  

Multiple Choice

Use the product rule to find the derivative. $$f\left(x\right)\ =\ -x^3\left(3x^4-2\right)$$  

• $$-3x^2+12x^3$$  

• $$-21x^3+6x$$  

• $$-21x^6+6x^2$$  

• $$-84x^6-6x^2$$  

Multiple Choice

$$x^2\left(-3x^2-2\right)$$  

• $$-6x^3-4x$$  

• $$-12x^3-4x$$  

• $$12x^3+4x$$  

• $$-36x^2-4$$  

Multiple Choice

Find the derivative of $$f\left(x\right)=\left(-x^3-2\right)\left(-4x^2-1\right)$$ . 

• $$12x^4+3x^2$$  

• $$20x^4+3x^2+16x$$  

• $$3x^5+32x^3+6x^2+8x$$  

• $$24x^3$$  

Multiple Choice

$$If\ y=\frac{3}{4+x^2},\ then\ \frac{dy}{dx}=$$  

• $$\frac{3}{2x}$$  

• $$\frac{3x}{\left(1+x^2\right)^2}$$  

• $$\frac{6x}{\left(4+x^2\right)^2}$$  

• $$-\frac{6x}{\left(4+x^2\right)^2}$$  

• $$-\frac{3}{\left(4+x^2\right)^2}$$  

Multiple Choice

Given that      $$y=\frac{x+2}{x-3}$$  

Find y'

• $$y'=-\frac{1}{\left(x-3\right)^2}$$  

• $$y'=-\frac{5}{\left(x-3\right)^2}$$  

• $$y'=\frac{2x-5}{\left(x-3\right)^2}$$  

• $$y'=\frac{2x-1}{\left(x-3\right)^2}$$

Multiple Choice

$$y\left(x\right)=\frac{4x^2+3}{2x^2-5}$$  

Find the derivative above.

• $$y\ '=\frac{-52x}{\left(2x^2-5\right)^2}$$  

• $$y\ '=\frac{-28x}{\left(2x^2-5\right)^2}$$

• $$y\ '=\frac{32x^3-52x+15}{\left(2x^2-5\right)^2}$$  

• $$y\ '=\frac{-10x}{\left(2x^2-5\right)^2}$$  

Multiple Choice

If $$y=\frac{4x^3}{2x^2-5}$$ , find $$\frac{dy}{dx}$$ . 

• $$8x^4-60x^2$$  

• $$16x^4$$  

• $$2-\frac{12}{5}x^2$$  

• $$\frac{8x^4-60x^2}{\left(2x^2-5\right)^2}$$  

Multiple Choice

Find the equation of the tangent line at the point (-1,1) of: $$f\left(x\right)\ =\ x^4$$  

• $$y=-\frac{1}{4}x-3$$  

• $$y=-4x-3$$  

• $$y=\frac{1}{4}x+3$$  

• $$y=4x+3$$  

Multiple Choice

What is the equation of the tangent line at x = 1  $$of$$   $$f\left(x\right)=\ \sqrt{x}$$  

• $$y=\frac{1}{2}x-\frac{1}{2}$$  

• $$y=\frac{1}{2}x+\frac{1}{2}$$  

• $$y=2x-1$$  

• $$y=2x$$  

Multiple Select

What is $$\frac{\text{d}}{\text{d}x}\left(\sqrt{x}-\frac{1}{x}\right)=?$$  . You can choose more than ONE answer.

• $$\frac{1}{2\sqrt{x}}-\frac{1}{x^2}$$  

• $$\frac{1}{2}\sqrt{x}+\frac{1}{x^2}$$  

• $$\frac{1}{2\sqrt{x}}+\frac{1}{x^2}$$  

• $$\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{x^2}$$  

Multiple Choice

Find the derivative of $$f\left(x\right)=4x^5+2x^3+7$$  

• $$f'\left(x\right)=20x^4+6x^2+7$$  

• $$f'\left(x\right)=20x^4+6x^2$$  

• $$f'\left(x\right)=20x^5+6x^3+7$$  

• $$f'\left(x\right)=20x^5+6x^3$$  

Multiple Choice

Find the derivative of $$f\left(x\right)=9x^4-50$$  

• $$f'\left(x\right)=36x^4-50$$  

• $$f'\left(x\right)=36x^4$$  

• $$f'\left(x\right)=36x^3-50$$  

• $$f'\left(x\right)=36x^3$$  

Multiple Choice

Find the derivative of $$y=\frac{4}{x^5}-9x$$  

• $$y'=-20x^{-6}-9$$  

• $$y'=20x^4-9$$  

• $$y'=-20x^{-4}-9$$  

• $$y'=-20x^{-6}-9x^{-1}$$  

Multiple Choice

Find the derivative of $$h\left(x\right)=x^{\frac{3}{4}}-\sqrt[7]{x}$$  

• $$h'\left(x\right)=\frac{3}{4}x^{-\frac{1}{4}}+7x^{-8}$$  

• $$h'\left(x\right)=\frac{3}{4}x^{-\frac{3}{4}}-\frac{1}{7}x^{-\frac{1}{7}}$$  

• $$h'\left(x\right)=\frac{3}{4}x^{-\frac{1}{4}}-\frac{1}{7}x^{-\frac{6}{7}}$$  

• $$h'\left(x\right)=\frac{3}{4}x^{-\frac{1}{4}}+\frac{1}{7}x^{-\frac{8}{7}}$$  

Multiple Choice

Find the derivative of $$y=\frac{9}{4x^8}$$  

• $$y'=\frac{72}{4}x^7$$  

• $$y'=\frac{1}{32x^7}$$  

• $$y'=\frac{9}{32}x^7$$  

• $$y'=\frac{-72}{4}x^{-9}$$  

Multiple Choice

Find the derivative of $$g\left(x\right)=x^{\frac{2}{5}}+\sqrt[]{x}$$  

• $$g'\left(x\right)=\frac{2}{5}x^{-\frac{2}{5}}+\frac{1}{2}x^{-\frac{1}{2}}$$  

• $$g'\left(x\right)=\frac{2}{5}x^{-\frac{3}{5}}-x^{-2}$$  

• $$g'\left(x\right)=\frac{2}{5}x^{-\frac{3}{5}}+\frac{1}{2}x^{-\frac{1}{2}}$$  

• $$g'\left(x\right)=\frac{2}{5}x^{-\frac{2}{5}}-x^{-2}$$  

Fill in the Blank

Suppose $$f\left(x\right)=\frac{8}{x^2}$$  . What is the value of $$f'\left(2\right)$$  

-

Fill in the Blank

Suppose $$f\left(x\right)=100\ \sqrt[]{x}$$  . What is the value of $$f'\left(25\right)$$  ?

Multiple Choice

Find the derivative of

$$g\left(x\right)=\left(x^2+1\right)^5$$  

• $$g'\left(x\right)=5\left(x^2+1\right)^4+2x$$  

• $$g'\left(x\right)=5\left(x^2+1\right)^4$$  

• $$g'\left(x\right)=5\left(2x\right)^4$$  

• $$g'\left(x\right)=10x\left(x^2+1\right)^4$$  

Multiple Choice

Differentiate $$\left(x+3\right)^5$$  

• $$5\left(x+3\right)^4$$  

• $$15\left(x+3\right)^4$$  

• $$5x\left(x+3\right)^4$$  

• $$15x\left(x+3\right)^4$$  

Multiple Choice

Differentiate $$\left(2x-1\right)^3$$  

• $$3\left(2x-1\right)^2$$  

• $$6\left(2x-1\right)^2$$  

• $$6x\left(2x-1\right)^2$$  

• $$2\left(2x-1\right)^3$$  

Multiple Choice

Differentiate $$\left(8-x\right)^7$$  

• $$7\left(8-x\right)^6$$  

• $$-7\left(8-x\right)^6$$  

• $$-7x\left(8-x\right)^6$$  

• $$7\left(8-x\right)^7$$  

Multiple Choice

Differentiate $$2\left(3x+4\right)^6$$  

• $$36\left(3x+4\right)^5$$  

• $$6\left(3x+4\right)^5$$  

• $$12\left(3x+4\right)^5$$  

• $$12x\left(3x+4\right)^5$$  

Multiple Choice

Differentiate $$\left(6-5x\right)^4$$  

• $$20\left(6-5x\right)^3$$  

• $$4\left(6-5x\right)^3$$  

• $$-20\left(6-5x\right)^3$$  

• $$-5\left(6-5x\right)^3$$  

Multiple Choice

Differentiate $$\sqrt[]{4x-1}$$  

• $$2(4x−1)^{-\frac{1}{2}}$$  

• $$2(4x−1)^{\frac{1}{2}}$$  

• $$4(4x−1)^{-\frac{1}{2}}$$  

• $$4(4x−1)^{\frac{1}{2}}$$  

Multiple Choice

$$\frac{dy}{dx}\ if\ y=\left(2x^2+4\right)^3$$  Find

• $$\left(12x+4\right)\left(2x^2+4\right)^2$$  

• $$3\left(2x^2+4\right)^2$$  

• $$12x\left(2x^2+4\right)^2$$  

• $$12\left(2x^2+4\right)^2$$  

Multiple Choice

Given  $$F\left(x\right)=\left(\frac{x}{2x+1}\right)^3$$ ,    find  $$F'\left(x\right)$$ . 

• $$3\left(\frac{x}{2x+1}\right)^2$$  

• $$3\left[\frac{1}{\left(2x+1\right)^2}\right]^2$$  

• $$\frac{1}{2}$$  

• $$\frac{3x^2}{\left(2x+1\right)^4}$$  

Match

Match the following

• $$\frac{d}{dx}\left(12\right)=$$  

• $$\frac{d}{dx}\left(12x\right)=$$  

• $$\frac{d}{dx}\left(x^{12}\right)=$$  

• $$\frac{d}{dx}\left(\frac{1}{12}x^{12}\right)=$$  

• $$\frac{d}{dx}\left(12x^2+2x\right)=$$  

0

12

12x¹¹

x¹¹

24x+2

• 0

• 12

• 12x¹¹

• x¹¹

• 24x+2

Multiple Choice

Find $$y'$$ if  $$y=-3\sqrt{1-8x}$$   

• $$y'=\frac{12}{\sqrt{1-8x}}$$  

• $$y'=\frac{-12}{\sqrt{1-8x}}$$  

• $$y'=\frac{3}{2\sqrt{1-8x}}$$  

• $$y'=\frac{-3}{2\sqrt{1-8x}}$$  

Multiple Choice

Differentiate:
f(x) = x⁷ (5 + 8x)³

• f '(x) = x² (5 + 8x)⁶ (35 + 80x)
• f '(x) = x⁶ (5 + 8x)² (35 + 80x)
• f '(x) = 8x⁷ (5 + 8x)² (35 + 80x)
• f '(x) = x⁶ (5 + 8x)³ (35 + 80x)

Multiple Choice

The UAE Hope Probe traveled more than 493 million kilometers in space over the course of seven months on its journey to Mars on February 9, 2021, making the UAE the fifth country to reach the Red Planet, celebrating the fiftieth anniversary of the establishment of the UAE according to the relationship

$$f\left(x\right)=\sqrt[3]{x}$$ , Find the slope of that track at x=1.

•   $$\frac{1}{3}$$

•   $$\frac{2}{7}$$

• $$\frac{1}{16}$$

• $$\frac{3}{11}$$