Final Exam - Final Revision T2-25

Final Exam - Final Revision

By Sahel Otoom

​Derivative Of Trigonometric Functions

Multiple Choice

Find y' if y = tan(3x²+2).

• sec²(3x²+2)
• 6xsec²(3x²+2)
• 6xsec²x(3x²+2)
• sec²(6x)

Multiple Choice

$$\frac{dy}{dt}\ if\ y=\cos\sqrt{10t+12}$$  Find

• $$\frac{-5}{\sqrt{10t+12}}\sin\sqrt{10t+12}$$  

• $$\frac{-1}{2\sqrt{10t+12}}\sin\sqrt{10t+12}$$  

• $$-\sin\sqrt{10t+12}$$  

• $$-\sin\left(\frac{5}{\sqrt{10t+12}}\right)$$  

Multiple Choice

Find the derivative: $$\sin^2\left(3x+2\right)$$  

• $$6\sin\left(3x+2\right)\cos\left(3x+2\right)$$  

• $$-6\sin\left(3x+2\right)\cos\left(3x+2\right)$$  

• $$6\cos\left(3x+2\right)$$  

• $$-6\cos\left(3x+2\right)$$  

Match

Match the following

• $$\frac{d}{dx}\left(\cos2x\right)$$

• $$\frac{d}{dx}\left(\tan3x\right)$$

• $$\frac{d}{dx}\left(\sec8x\right)$$

• $$\frac{d}{dx}\left(\sin x\right)$$

• $$\frac{d}{dx}\left(\cot x\right)$$

- 2sin 2x

3sec² 3x

8sec8x tan8x

cos x

-csc²x

• - 2sin 2x

• 3sec² 3x

• 8sec8x tan8x

• cos x

• -csc²x

Multiple Choice

Emsat

Find $$\frac{\text{d}}{\text{d}x}\left(\ 4\sin\ 6x\right)=$$  

• 24 cos x

• 24 sin 6x

• 24 cos 6x

• -24 cos 6x

Multiple Choice

If $$y\ =\ 3x\ -\ 5\ \cos\left(2x^2-3\right)$$  . Find $$\frac{\text{d}y}{\text{d}x}$$  .

• $$3+\ 20x\ \sin\ \left(2x^2-3\right)$$  

• $$3-20x\ \sin\left(2x^2-3\right)$$   

• $$3\ -\ 20\ \sin\ \left(2x^2-3\right)$$  

• $$3\ +\ 20\ \sin\left(2x^2-3\right)$$  

Multiple Choice

Derive $$y\ =\ 4x\ \tan\ 6x.$$  

• $$4x\ \tan\ 6x\ +\ 24\ \sec^26x$$  

• $$4x\ \tan\ 6x\ +\ 24x\ \csc^26x$$  

• $$4\ \tan\ 6x\ +\ 24\ x\ \sec^26x$$  

• $$4\tan6x\ -\ 24x\ \csc^26x$$  

Math Response

Find the slope of the tangent line to

$$y=\tan\ 3x\ at\ a=0$$ .

Type answer here

Correct Answer: 3

Deg^°

Rad

Derivatives of Exponential and Logarithmic Functions

Introduction to Calculus

Multiple Choice

$$Find\ \frac{dy}{dx}\ if\ y=e^x\tan x\ .$$

• $$e^x\sec^2x+e^x\tan x$$

• $$e^x\sec^2x+e^x\tan^2x$$

• $$e^x\sec^2x-e^x\tan x$$

• $$e^x\sec^2x$$

Multiple Choice

If $$y=e^{-2x^3}$$  , then  $$\frac{\text{d}y}{\text{d}x}=$$  

• $$-2x^3e^{-2x^3}$$  

• $$e^{-2x^3}$$  

• $$-6x^2e^{-2x^3}$$  

• $$e^{-6x^2}$$  

• $$-6x^2e^{-6x^2}$$  

Multiple Choice

$$y=e^{4x}?$$  Find the derivative of

• $$e^{4x}$$  

• $$\frac{e^{4x}}{4}$$  

• $$4e^{4x}$$  

• $$-4e^{4x}$$  

Multiple Choice

The exact value of $$f'\left(-2\right)$$  if  $$f\left(x\right)=2e^{-3x}$$  is:

• $$-6e^6$$  

• $$6e^6$$  

• $$-6e^{-6}$$  

• $$6e^{-6}$$  

• $$-e^6$$  

Math Response

What is the derivative of

$$y=\ln\ x^2$$

Type answer here

Correct Answer: \frac{2}{x}

Deg^°

Rad

Dropdown

What is the derivative of

$$y=\ln\cos x$$ ​

Select answer

Math Response

$$find\ the\ derivative\ of\ y=\ln\left(x^2+x\right)$$

Type answer here

Correct Answer: \frac{2x+1}{x^2+x}

Deg^°

Rad

Drag and Drop

$$find\ the\ derivative\ of\ y=\ln\left(\sin x\right)$$ ​

Placeholder

Drag these tiles and drop them in the correct blank above

cot x

tan x

sinx

ln (cos x)

None

Math Response

$$find\ the\ derivative\ of\ y=\ln e^x$$

Type answer here

Correct Answer: 1

Deg^°

Rad

Multiple Choice

$$find\ the\ derivative\ of\ y=\ln\left(x^2+x\right)$$  

• $$\frac{\text{d}x}{\text{d}y}=\frac{1}{2x+1}$$  

• $$\frac{\text{d}x}{\text{d}y}=\frac{x^2+x}{2x+1}$$  

• $$\frac{\text{d}x}{\text{d}y}=\frac{2x+1}{x^2+x}$$  

• $$\frac{\text{d}x}{\text{d}y}=x^2+x$$  

Multiple Choice

What is the derivative of ln(x)?

• xln(x)

• $$\frac{1}{x}$$

• $$e^x$$

• ln(x+1)

Multiple Choice

Differentiating  $$-\ln\left(2x-3\right)$$  yields:

• $$\frac{2}{2x-3}$$  

• $$\frac{-1}{2x-3}$$  

• $$\frac{1}{2x-3}$$  

• $$-\frac{2}{2x-3}$$  

• $$\frac{-3}{2x-3}$$  

Multiple Choice

Differentiating  $$e^{-x}.\ln\left(x+3\right)$$  yields:

• $$\frac{e^{-x}}{x+3}-e^{-x}.\ln\left(x+3\right)$$  

• $$\frac{e^{-x}}{x+3}+e^{-x}.\ln_{ }\left(x+3\right)$$  

• $$e^{-x}\left(x+3\right)-e^{-x}.\ln_{ }\left(x+3\right)$$  

• $$e^{-x}\left(x+3\right)+e^{-x}.\ln_{ }\left(x+3\right)$$  

Multiple Choice

$$find\ the\ derivative\ of\ f(x)=e^{x2}$$  

• $$\frac{\text{d}x}{\text{d}y}=x^2e^{x^2}$$  

• $$\frac{\text{d}x}{\text{d}y}=2xe^{x^2}$$  

• $$\frac{\text{d}x}{\text{d}y}=2x+e^{x^2}$$  

• $$\frac{\text{d}x}{\text{d}y}=e^{2x}$$  

Multiple Choice

$$find\ the\ derivative\ of\ f\left(x\right)=xe^x$$  

• $$\frac{\text{d}x}{\text{d}y}=xe^x+e^x$$  

• $$\frac{\text{d}x}{\text{d}y}=e^x$$  

• $$\frac{\text{d}x}{\text{d}y}=1+e^x$$  

• $$\frac{\text{d}x}{\text{d}y}=x+e^x$$  

Implicit Differentiation

by Sahel Otoom

Multiple Choice

The traditional Arabic coffee pot (dallah) design follows the curve $$x^2y+xy^2=25$$ , where x and y are in centimeters. Find $$\frac{dy}{dx}$$ .

• $$\frac{-2xy-y^2}{x^2+2xy}$$

• $$\frac{-2xy-4y^2}{x^2+2xy}$$

• $$\frac{-2xy+y^2}{x^2+2xy}$$

• $$\frac{2xy-y^2}{x^2+2xy}$$

Multiple Choice

Which step below is the first incorrect step to find the slope of the line tangent to the curve 𝑥³ + 𝑦³ − 9𝑥𝑦 = 0 at point (2, 4). And correct the error.

Step 1 : 3𝑥² + 3𝑦²𝑦 ′ − 9 (𝑥𝑦 ′ + 𝑦) = 0

Step 2 : 3𝑥² + 3𝑦²𝑦 ′ − 9𝑥𝑦 ′ − 9𝑦 = 0

Step 3 : 𝑦 ′ (3𝑦² − 9𝑥) = 9𝑦 + 3𝑥²

Step 4: slope= 4/5

• Step 1

  3𝑥² + 3𝑦²𝑦 ′ + 9 (𝑥𝑦 ′ + 𝑦) = 0

• Step 2

  3𝑥² + 3𝑦²𝑦 ′ − 9𝑥𝑦 ′ + 9𝑦 = 0

• Step 3

  𝑦 ′ (3𝑦² − 9𝑥) = 9𝑦 -3𝑥²

• Step 4

  slope= 4/7

Multiple Choice

Differentiate w.r.t.x

$$y\ ^7$$  

• $$7y\ ^6\times\frac{\text{d}y}{\text{d}x}$$  

• $$8y\ ^7$$  

• $$7y^6$$  

• $$8y^{7\ }\times\frac{\text{d}y}{\text{d}x}$$  

Multiple Choice

Differentiate w.r.t.x

$$x^2+y^2+2$$  

• $$2x+2y\frac{\text{d}y}{\text{d}x}+2$$  

• $$2x+2y\frac{\text{d}y}{\text{d}x}$$  

• $$2x+2y$$  

Multiple Choice

Find  $$\frac{dy}{dx}$$ :  $$5x^2\ =3y^2+1$$  

• $$\frac{5x}{3y}$$  

• $$\frac{3y}{5x}$$  

• $$\frac{4x}{3y}$$  

• $$\frac{2y}{3x}$$  

Multiple Choice

Find  $$\frac{dy}{dx}$$ :  $$y^2=10x$$  

• $$\frac{5}{y}$$  

• $$\frac{10}{y}$$  

• $$\frac{y^2}{5}$$  

• $$\frac{y^2}{10}$$  

Multiple Choice

Find the derivative of $$x^2+xy+y^3=0$$  

• $$-\frac{x+3y^2}{2x+y}$$  

• $$-\frac{2x+y}{x+3y^2}$$  

• $$-\frac{2x}{x+3y^2}$$  

Multiple Choice

Find dy/dx at the given point

x³ +2xy -y²=11 at (2,3)

• -4/7

• 12

• -9

• 9

Multiple Choice

Given $$y^4-4xy+50=0$$  . What is $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{x}{y^3}$$  

• $$\frac{\text{d}y}{\text{d}x}=\frac{y}{y^3-x}$$  

• $$\frac{dy}{dx}=\frac{-y}{y^3-x}$$  

​Increasing and Decreasing Functions

Multiple Select

Find all the critical numbers of

f(x) = 2x³ – 6x² + 12 on the interval on [-2,3]

• x= 4

• x= 0 and x=2

• x= -2 and x=9

• x= 2

Multiple Select

$$f\left(x\right)=\frac{2}{3}x^3-6x^2+16x+14$$   Find the critical numbers of f(x) on the interval on [1,8]

• x=2 and x=4

• x=0 and x=8

• x = 40

Multiple Choice

Find the critical number of f(x) = 4x² – 8x + 1

• x = 2

• x = 0

• x = 8

• x = 1

Multiple Select

$$f\left(x\right)=\frac{1}{3}x^3+3x^2+8x-5$$   Find the critical numbers of f(x).   Select ALL that apply

• x=2

• x=0

• x= -4

• x= -2

Multiple Select

Find all the critical numbers of f(x).

$$f\left(x\right)=\frac{1}{4}x^4-\frac{5}{3}x^3-7x^2+13$$   Select ALL that apply.

• x= 7

• x= 5

• x= 0

• x = -2

• x = 14

Multiple Choice

The function f(x) has a derivative f' (x) which is always negative. What can be said about f(x)?

• It is always increasing

• It is always decreasing

• It has a local maximum

• It has a local minimum

Multiple Choice

A point where a function's derivative is zero or undefined is called a _____________.

• inflection point

• critical point

Multiple Choice

List the types of intervals that the graph demonstrates in order.

• decreasing, constant, increasing, constant
• increasing, decreasing, constant
• increasing, constant, increasing, constant
• increasing, constant, decreasing, constant

Multiple Choice

What is the increasing interval on the function shown?

• $$\left(-\infty,\ 1\right)$$

• $$\left(-\infty,\ 2\right)$$

• $$\left(2,\ \infty\right)$$

• $$\left(1,\ \infty\right)$$

Multiple Choice

What is the decreasing interval on the function shown?

• $$\left(-\infty,\ 1\right)$$

• $$\left(-\infty,\ 2\right)$$

• $$\left(2,\ \infty\right)$$

• $$\left(1,\ \infty\right)$$

Multiple Choice

What is the increasing interval on the function shown?

• $$\left(-\infty,\ -3\right)$$

• $$\left(-\infty,\ -4\right)$$

• $$\left(-4,\ \infty\right)$$

• $$\left(-3,\ \infty\right)$$

Multiple Choice

What is the decreasing interval on the function shown?

• $$\left(-\infty,\ -3\right)$$

• $$\left(-\infty,\ -4\right)$$

• $$\left(-4,\ \infty\right)$$

• $$\left(-3,\ \infty\right)$$

Multiple Choice

Over what interval is this function constant?

• (-3,5)

• (-3, 4)

• (4,8)

• -5

Multiple Choice

in interval A the function is

• increasing

• decreasing

• constant

Multiple Choice

Find the point(s) of inflection (if it exists) of the function:

$$f\left(x\right)=-x^2+7x-10$$   Select ALL the correct answer(s).

• No point of inflection 

• At x = 2

• At x = 5

• At  x = $$\infty$$  

Multiple Choice

in interval B the function is

• increasing

• decreasing

• constant

Multiple Choice

in interval C the function is

• increasing

• decreasing

• constant

Multiple Choice

in interval E the function is

• increasing

• decreasing

• constant

Multiple Choice

If the derivative of a function is always positive, the function is always _____________

• Increasing

• Decreasing

Multiple Choice

If the derivative of a function is always ______________, the function is always decreasing

• Positive

• negative

Multiple Select

Given      $$f\left(x\right)=-4x^2+24x+5$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. Select ALL that apply.

• Increasing at   $$\left(-\infty,\ 3\right)$$  

• Decreasing at $$\left(-\infty,\ 3\right)$$  

• Decreasing at $$\left(3,\ \infty\right)$$  

• Increasing at    $$\left(3,\infty\right)$$  

Multiple Select

Given      $$f\left(x\right)=3x^2+12x+15$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. Select ALL that apply.

• Increasing at   $$\left(-\infty,-2\right)$$  

• Decreasing at $$\left(-\infty,\ -2\right)$$  

• Decreasing at $$\left(-2,\ \infty\right)$$  

• Increasing at    $$\left(-2,\infty\right)$$  

Multiple Choice

When a function changes from increasing to decreasing, a ___________ occurs.

• Relative Maximum

• Relative Minimum

• Point of Inflection

• Frown

Multiple Select

Given      $$f\left(x\right)=\frac{5}{2}x^2-30x+20$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. Select ALL that apply.

• Increasing at   $$\left(-\infty,6\right)$$  

• Decreasing at $$\left(-\infty,\ 6\right)$$  

• Decreasing at $$\left(-6,\ \infty\right)$$  

• Increasing at    $$\left(6,\infty\right)$$  

Multiple Choice

When a function changes from decreasing to increasing, a ___________ occurs.

• Relative Maximum

• Relative Minimum

• Point of Inflection

• Smile

Multiple Select

Given      $$f\left(x\right)=-3x^2-6$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. Select ALL that apply.

• Decreasing at   $$\left(-\infty,-1\right)$$  

• Increasing at $$\left(-\infty,\ 0\right)$$  

• Decreasing at $$\left(0,\ \infty\right)$$  

• Increasing at    $$\left(-1,\infty\right)$$  

Multiple Select

Given      $$f\left(x\right)=2x^3-9x^2-60x+40$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. Select ALL that apply.

• Decreasing at  ( -2,  5)

• Decreasing at  $$\left(-\infty,\ -2\right)$$  

• Increasing  at $$\left(-\infty,\ -2\right)$$  

• Increasing at    $$\left(5,\infty\right)$$  

Multiple Select

Given      $$f\left(x\right)=-x^3+48x$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. 

Select ALL the correct answers.

• Decreasing at  ( -4, 4)

• Decreasing at  $$\left(-\infty,\ 4\right)$$  

• Increasing at     (-4,  4)

• Increasing at    $$\left(4,\ \infty\right)$$  

Multiple Select

Given      $$f\left(x\right)=-2x^3+9x^2+60x+10$$  

Use the First Derivative Test to find the open interval in which f(x) is increasing or decreasing. Select ALL that apply.

• Increasing at    $$\left(-2,\ 5\right)$$  

• Increasing at  $$\left(-\infty,\ -2\right)$$  

• Decreasing  at $$\left(-\infty,\ -2\right)$$  

• Decreasing at    $$\left(5,\infty\right)$$  

Multiple Choice

Use the sign chart for f'(x).  There is(are) ...

• a local maximum at x = -2.
• a local minimum at
  x = -2 and a local maximum at x = 4.
• a local maximum at
  x = -2 and a local minimum at x = 4.
• no extrema.

Multiple Choice

Use the sign chart for f'(x).
There is(are) ...

• a local maximum at x = -2.
• a local maximum at x = 4.
• local maxima at x = -2 and x = 4.
• no extrema.

​Concavity And Second Derivative Test

Multiple Choice

If f'' (x)>0, what can be said about the function f(x)?

• It is concave up

• It is increasing

• It is decreasing

• It has a maximum

Multiple Choice

The function p(x)=x²-6x+9

• Always concave up

• Always concave down

• Concave up at x<3 , concave down at x>3

• Neither concave up nor down

Labelling

Fill in the Blank: Fill in the blank with the correct words.

Drag labels to their correct position on the image

negative

concave up

second

first

inflection

Multiple Choice

Find the second derivative of the polynomial...

• 12x + 6

• 6x² + 6x

• 12x + 6x

• 2x² + 6x

Multiple Choice

For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3.

• increasing
• decreasing
• concave up
• concave down

Multiple Choice

The concavity of a function is described by its _______________.

• first derivative
• second derivative
• third derivative
• expression

Multiple Choice

What will be true at an inflection point?  (select the best answer)

• f(x)=0
• f'(x)=0
• f''(x)=0
• The function is undefined

Multiple Choice

Where is the point of inflection for the function  $$f\left(x\right)=x^3+6x^2$$  ?

• $$x=0$$  

• $$x=-4$$  

• $$x=-2$$  

• $$x=2$$  

Multiple Choice

Discuss the concavity and the point(s) of inflection (if any) of the function:

$$f\left(x\right)=x^3-3x^2+10$$  

• Concave down in $$\left(-\infty,1\right)$$  , Concave up in . And the point of inflection at x = 1

•  Concave down in  $$\left(-\infty,\infty\right)$$ . with no point of inflection.

• Concave up in  $$\left(1,\infty\right)$$ . with no point of inflection.

Multiple Choice

Discuss the concavity and the point(s) of inflection (if any) of the function:

$$f\left(x\right)=-2x^3-24x^2-14x+5$$

• Concave up in $$\left(-\infty,-4\right)$$ Concave down in , And the point of inflection at x = - 4

• Concave down in $$\left(-\infty,-4\right)$$ Concave up in , And the point of inflection at x = - 4

• Concave up in $$\left(-\infty,\infty\right)$$ with No point of inflection

Multiple Choice

Discuss the concavity and point(s) of inflection (if any) for the function: 

$$f\left(x\right)=x^4-24x^2+11x+40$$

• Concave up in $$\left(-\infty,-2\right)$$ and , Concave down in (-2, 2). And points of inflection at x = -2 and 2

• Concave down in $$\left(-2,\ 2\right)$$  and , Concave up in (-2, 2). And points of inflection at x = -2 and 2

• Concave down in  $$\left(4,\infty\right)$$ ,With no points of inflection.

Multiple Choice

Discuss the concavity and the point(s) of inflection (if any) for the function

$$f\left(x\right)=-2x^5-5x+2$$  

• Concave down in  $$\left(0,\infty\right)$$  Concave up in ,. And Point of Inflection at x = 0

• Concave up in  $$\left(-\infty,0\right)$$  Concave down in ,. And Point of Inflection at x = 0

• Concave down in $$\left(-\infty,0\right)$$  and  . with no inflection point.

• Concave up in $$\left(0,\infty\right)$$  and  . with no inflection point.

Labelling

Drag and drop

Drag labels to their correct position on the image

Multiple Choice

The function p(x)=x²-6x+9 is

• Always concave up

• Always concave down

• Concave up at x<3 , concave down at x>3

• Neither concave up nor down

Multiple Choice

The function P(t)=0.1t³-0.6t²+t+10 represents the population growth (in millions) in Abu Dhabi. Find all inflection points.

• t=2

• t=3

• t=4

• t=5

Multiple Choice

Explain how the second derivative test helps determine the concavity of a function.

• If f''(x) > 0, f(x) has a local minimum at that point.

  If f''(x) < 0, f(x) has a local maximum at that point.

• If f''(x) > 0, f(x) has a local maximum at that point.

  If f''(x) < 0, f(x) has a local minimum at that point.

• If f''(x) > 0, f(x) has a local minimum at that point.

  If f''(x) < 0, f(x) has a local minimum at that point.

• If f''(x) > 0, f(x) has a local maximum at that point.

  If f''(x) < 0, f(x) has a local maximum at that point.

Multiple Choice

On what interval(s) is the function  $$f\left(x\right)=x^3+6x^2$$ concave down? 

• $$\left(-\infty,-4\right)$$

• $$\left(-\infty,-2\right)$$  

• $$\left(-2,\infty\right)$$

• $$\left(0,\infty\right)$$

Multiple Choice

concavity

• concave up only

• concave down only

• both concave up and down

• none

Multiple Choice

For a function g(x), g''(3)=22 indicates that g(x) is ____________ at x=3.

• increasing
• decreasing
• concave up
• concave down

Multiple Choice

Find the point(s) of inflection (if it exists) of the function:

$$f\left(x\right)=-x^2+7x-10$$   Select ALL the correct answer(s).

• No point of inflection 

• At x = 2

• At x = 5

• At  x = $$\infty$$