Midterm Exam Final Revision

Midterm Exam Final Revision

By Sahel Otoom

The chain rule

by Sahel Otoom

Multiple Choice

Emsat

Given f(x) = 2x and g(x) = x²+3 , find f(g(x)).

• x²+2x+3
• 4x²+3
• 2x²+3
• 2x²+6

Multiple Choice

Given f(x) = 3x + 10
and g(x) = x - 2
Find f(g(5))

• 19
• 23
• -10
• None of these.

Multiple Choice

Find the derivative of  f(x) = (x⁶ + 4)⁵

• f '(x) = 5x⁵(x⁴ + 4)⁴
• f '(x) = 6x⁵(x⁶ + 4)⁴
• f '(x) = 30x⁵(x⁶ + 4)⁴
• f '(x) = 30x⁶(x⁶ + 4)⁴

Multiple Choice

Find the derivative of f(x)=(x³-2x)²

• 6x⁵ - 12x³+8x
• 6x⁵ - 16x³+8x
• x⁶-4x⁴+4x²
• 6x⁵ - 16x^3-8x

Multiple Choice

• True

• False

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: $$\sin^2\left(3x+2\right)$$  

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

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

• $$6\cos\left(3x+2\right)$$  

• $$-6\cos\left(3x+2\right)$$  

Poll

How do you feel about chain rule?

• not good

• good

• really good

• excellent!

Implicit Differentiation

by Sahel Otoom

Multiple Select

Which of the following equations, if any, are written in implicit form?

(Check all that apply)

• $$y=4\ x-9$$

• $$x^{\ 2}+y^{\ 2}=49$$

• $$x\ y=16$$

• $$y=\sqrt{36-x^{\ 2}}$$

• $$y=\frac{25\ x}{x^{\ 2}-64}$$

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

Find the slope of the tangent line 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 $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{x}{y^3}$$  

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

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

Multiple Choice

Given  $$y\sin x\ +4\sec\ x\ =4$$  . What is  $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{-y\cos x-4\sec x\tan x\ }{\sin\ x}$$  

• $$\frac{dy}{dx}=\frac{-4\sec x\tan x}{\cos x}$$  

Poll

Choose the picture that describe your feeling about today's lesson

Related Rates

By Sahel Otoom

Multiple Choice

Obtain the Related Rate with respect to t depend to equation

$$A=\pi r^2$$  ,where $$\pi=\frac{22}{7}$$  ?

• $$\frac{dA}{dt}=2\pi r\ \frac{dr}{dt}$$  

• $$\frac{dA}{dt}=\pi r\ \frac{dr}{dt}$$  

• $$A=2\pi r$$  

Multiple Choice

Based on the following steps, which is the first step to perform Related Rates?

• Collect all terms involving dy/dt on the left side of the equation and move all other terms to the right side of the equation

• Differentiate both sides of the equation with respect to t

• Factor dy/dt out of the left side of the equation

• Solve for dy/dt

​Remember

Sphere Volume​

​

​

Multiple Choice

A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 4 in?

• -262π in³/sec

• -247π in³/sec

• -256π in³/sec

• -263π in³/sec

Multiple Choice

A spherical snowball is rolled in fresh snow, causing it to grow so that its radius increases at a rate of 4 in/sec. How fast is the volume of the snowball increasing when the radius is 9 in?

• 1296π in³/sec

• 1294π in³/sec

• 1303π in³/sec

• 1305π in³/sec

​circle area

​circle circumference

Multiple Choice

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 5 m/min. How fast is the area of the spill increasing when the radius is 5 m?

• 50π m²/min

• 47π m²/min

• 52π m²/min

• 40π m²/min

Multiple Choice

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft? What rate is given in the problem?

• $$\frac{dC}{dt}=40$$

• $$\frac{dr}{dt}=40$$

• $$\frac{dA}{dt}=40$$

• $$\frac{dπ}{dt}=40$$

Multiple Choice

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What rate are we looking for and when?

• $$\frac{dA}{dt}\ when\ C=100π$$

• $$\frac{dC}{dt}\ when\ A\ =\ 100π$$

• $$\frac{dA}{dt}\ when\ A\ =\ 100π$$

• $$\frac{dC}{dt}\ when\ r\ =\ 100π$$

Multiple Choice

What is the derivative of area with respect to time?

• $$\frac{dA}{dt}=2πr\cdot\frac{dr}{dt}$$

• $$\frac{dA}{dt}=2πr$$

• $$\frac{dA}{dt}=πr\cdot\frac{dr}{dt}$$

• $$\frac{dA}{dt}=πr^2\cdot\frac{dr}{dt}$$

Multiple Choice

What is the derivative of circumference with respect to time?

• $$\frac{dC}{dt}=2π\cdot\frac{dr}{dt}$$

• $$\frac{dC}{dt}=2π$$

• $$\frac{dC}{dt}=2πr$$

• $$\frac{dC}{dt}=π\cdot\frac{dr}{dt}$$

Multiple Choice

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What is the length of the radius when the circumference is 100π ft?

• $$r=50\ ft$$

• $$r=100\ ft$$

• $$r=\sqrt{50}ft$$

• cannot be determined

Multiple Choice

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

At what rate is the radius changing when the circumference is 100π ft?

• $$\frac{dr}{dt}=\frac{20}{π}$$

• $$\frac{dr}{dt}=\frac{50}{π}$$

• $$\frac{dr}{dt}=50$$

• $$\frac{dr}{dt}=20$$  

Multiple Choice

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

• $$\frac{dA}{dt}=2000\ \frac{ft^2}{\sec}$$

• $$\frac{dA}{dt}=40\ \frac{ft^2}{\sec}$$

• $$\frac{dA}{dt}=100\ \frac{ft^2}{\sec}$$

• $$\frac{dA}{dt}=200\ \frac{ft^2}{\sec}$$  

Multiple Choice

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec.  How fast is the height of the water changing when the water is 2 ft deep?

• $$\frac{-9}{8\pi}$$   ft/sec

• -9 ft/sec

• -8 $$\pi$$   ft/sec

Multiple Choice

A tank is in the form of an inverted cone having an altitude of 10 ft and a radius of 5 feet. Water is flowing into the tank at the rate of 1 ft³/min. How fast is the water level rising when the water is 3 ft deep?

• 3.4 ft/min
• 0.14 ft/min

• 1 ft/min
• 1/2  ft/min

Multiple Choice

A cube is shrinking at a rate of  

$$27\ \frac{m^3}{\min}$$  .  At what rate are the sides of the cube changing when the sides are 9 meters each?

• $$-\frac{1}{9}\ \frac{m}{\min}$$  

• $$-9\ \frac{m}{\min}$$  

• $$-9\ \frac{m^2}{\min}$$  

• $$-\frac{1}{9}\ \frac{m^3}{\min}$$  

Multiple Choice

A spherical balloon is inflated so that its radius increases at a rate of 2 cm/sec. How fast is the VOLUME of the balloon changing when the radius is 6 cm? NOTE:

$$V=\frac{4}{3}\pi r^3$$  

• $$288\pi\ \frac{cm^3}{\sec}$$  

• $$-288\pi\ \frac{cm^3}{\sec}$$  

• $$288\pi\ \frac{cm^2}{\sec}$$  

• $$-288\pi\ \frac{cm^2}{\sec}$$  

Multiple Choice

The perimeter of a rectangle is expressed by the equation $$P=2w+2l$$  

 What is its "Related Rates derivative"?

• $$\frac{dP}{dt}=2\frac{dw}{dt}+2\frac{dl}{dt}$$  

• $$\frac{dP}{dt}=\frac{dw}{dt}+\frac{dl}{dt}$$  

• $$P'=2w'l+2wl'$$  

• $$\frac{dP}{dx}=2\frac{dw}{dx}+2\frac{dl}{dx}$$  

Multiple Choice

What is the "Related Rates derivative" of   $$a^2+b^2=4^2$$  ?

• $$2a\frac{da}{dt}+2b\frac{db}{dt}=0$$  

• $$2a\frac{da}{dt}+2b\frac{db}{dt}=8$$  

• $$2a+\frac{db}{dt}+2b+\frac{da}{dt}=16$$  

• $$\frac{dy}{dx}=2a\frac{da}{dt}+2b\frac{db}{dt}$$  

Multiple Choice

The area of a square is increasing at  $$5\ \frac{cm^2}{\sec}$$ .  At what rate is a side x changing when it is $$3\ cm$$  long?

Which is true?

• $$A=5$$  

• $$\frac{dA}{dt}=3$$  

• $$x=3$$  

• $$\frac{dx}{dt}=3$$  

Multiple Choice

The variables x and y are both differentiable functions of t and are related by the equation $$y=x^3+3x+8$$  . Find dy/dt when x = 2, given that dx/dt = 5 when x = 2.

• 75

• 88

• -45

Multiple Choice

Circular Hotel Swimming Pool Al Ain Abu Dhabi UAE. The radius of the pool decreases at a rate of 4 m/min. How fast is the area of the pool increasing when the radius is 8 m?

• -64π m²/min

• 80π m²/min

• 78π m²/min

Multiple Choice

The variables x and y are both differentiable functions of t and are related by the equation $$y=x^2+3$$  . Find dy/dt when x = 1, given that dx/dt = 2 when x = 1.

• 4

• -4

• -8

Poll

Choose the picture that describe your feeling about today's lesson

Extrema on an Interval

By Sahel Otoom

 

 

​

Multiple Choice

What is an Absolute Maximum / Minimum

• The highest/lowest point for the entire function/graph.

• Any peak in the function

• The part that tends to infinity or negative infinity of the graph

Multiple Choice

f(x) = x² + 1 has both a minimum and a maximum on the closed interval [–1, 2].

• True

• False

Multiple Choice

What is the absolute minimum

of this function?

• None since

  $$y\rightarrow\infty$$

• y=0

  
  

• y=4

• x=4

Multiple Choice

How many relative extrema are in the picture? (Remember that Absolute extrema are also relative)

• 2

• 3

• 4

Multiple Choice

What is the minimum value of the function graphed?

• (1,-1)

• (3,1)

• y = -1

Multiple Choice

The blue dot on this graph represents a(n)...

• Absolute Maximum

• Relative Maximum

• Relative Minimum

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 Select

Given  $$0\le x\le2\pi$$  , Find ALL the critical numbers in 

$$f\left(x\right)=\cos x$$  . Select ALL that apply.

• $$x=0$$  

• $$x=\pi$$  

• $$x=2\pi$$  

• $$x=\frac{3\pi}{2}$$  

• $$x=\frac{\pi}{2}$$  

Multiple Choice

Find the critical number of

$$f\left(x\right)=2e^x-2x$$  

• x = 2

• x = 0

• x = 1

• x = e

Multiple Choice

Find the extrema of f(x) = 3x³ – 3x + 1 on the interval [–2, 2].

• Absolute Max at

  (2 , 3) and Absolute Min at

  (-1 , -3)

• Absolute Max at

  (-2 , 3) and Absolute Min at

  (2 , -3)

• Absolute Max at

  (-2 , -1) and Absolute Min at

  (-1 , -1)

Multiple Choice

Find the absolute maximum for the function on the given interval.

f(x) = x³ + 6x² + 9x + 3 on [-4,0]

• -4

• -1

• 0

• 3

Multiple Choice

Find the absolute maximum for the function on the given interval.

f(x) = x³ + 6x² + 9x + 3 on [-4,0]

• (-3, 3) & (-1, -1)

• (-4, -1) & (-1, -1)

• x = - 3, - 1

• (-3, 3) & (0, 3)

Multiple Choice

If (a,b) is a local minimum, then what will be true about f '(a)?

• It's positive

• It's negative

• It's zero

• Cannot be determined

Multiple Choice

Find the absolute extrema of

g(x) = 3x³ + 6x² on [-1, 1].

• abs max value of $$\frac{32}{9}$$ when x = $$-\frac{4}{3}$$ abs min value of 0 when x = 0

• abs max value of 9 when x = 1, abs min value of 3 when x = -1

• abs max value of 9 when x = 1, abs min value of 0 when x = 0

• abs max value of 9 when x = 1, abs min of -1 when x = -1

Poll

Choose the picture that describe your feeling about today's lesson