Algebra and Differential Calculus for Data Science Coursera Quiz Answers

Get Algebra and Differential Calculus for Data Science Coursera Quiz Answers

Are you interested in Data Science but lack the math background for it? Has math always been a tough subject that you tend to avoid? This course will teach you the most fundamental Calculus concepts that you will need for a career in Data Science without a ton of unnecessary proofs and techniques that you may never use.

Consider this an expressway to Data Science with approachable methods and friendly concepts that will guide you to truly understanding the most important ideas in Differential Calculus. We will review some algebra basics, talk about what a derivative is, compute some simple derivatives and apply the basics of derivatives to graphing and maximizing functions.

This course is designed to prepare learners to successfully complete Statistical Modeling for Data Science Application, which is part of CU Boulder’s Master of Science in Data Science (MS-DS) program.

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Week 1: Functions and Algebra Review

Quiz 1: Functions Quiz

Q1. Is the following graph a function of x ?

• Yes
• No

Q2. Is the following graph a function of x?

• Yes
• No

Q3. Is the following graph a function of x?

• Yes
• No

Q4. Is the following graph a function of x?

• Yes
• No

Q5. Is the following graph a function of x?

• Yes
• No

Q6. Compute f(0), f(3)f(0),f(3) and f(-2)f(−2) for f(x) = \frac {x-3}{x+2}f(x)=x+2x−3​

f(0) = f(0)=_____ , f(3) =f(3)= ______ , f(-2) =f(−2)= _______

Fill in the blanks and separate with a comma, ie. 1, 2, 3. If an answer is undefined then write undefined.

Answer: 

Q7. Compute f(0), f(4),f(0),f(4), and f(-2)f(−2) to 3 decimal places for f(t) = 4 e^{-2t}f(t)=4e−2t f(0) = f(0)=______ , f(4) =f(4)= ______ , f(-2)=f(−2)= ______

Fill in the blanks and separate with a comma, ie. 1, 2, 3. If an answer is undefined then write undefined.

Answer: 

Q8. Compute f(0), f(10),f(0),f(10), and f(-10)f(−10) for f(x) = \sqrt{x+6}f(x)=x+6​ (up to 3 decimal places if needed)

f(0) = f(0)=_______, f(10) =f(10)= _______ , f(-10) = f(−10)=_______

Fill in the blanks and separate with a comma, ie. 1, 2, 3. If an answer is undefined then write undefined.

Answer: 

Quiz 2: Domain and Range Quiz

Q1. W​hat is the domain and range of the following function?

• D​omain: all real numbers, Range: all real numbers
• D​omain: x>=-1, Range: all real numbers
• D​omain: all real numbers, Range: y>=-1
• D​omain: x>=-1, Range: y>=-1

Q2. W​hat is the domain and range of the following function?

• D​omain: all Real numbers, Range: all Real numbers
• D​omain: x=2, Range: all Real numbers
• D​omain: all Real numbers, Range: y=2
• D​omain: x=2, Range: y=2

Q3. W​hat is the domain and range of the following function?

• D​omain: all Real numbers, Range: all Real numbers
• D​omain: x<=3, Range: y>=-1
• D​omain: all Real numbers, Range: y>=-1
• D​omain: x<=3, Range: all Real numbers

Quiz 3: Piecewise Functions Quiz

Q1. C​ompute f(0) using the piecewise function showing below, refer to this figure as Figure A:

Answer: 

Q2. U​sing figure A., what is f(-5)?

Answer: 

Q3. U​sing Figure A., what is f(1)?

Answer: 

Q4. U​sing Figure A., what is f(2)?

Answer: 

Q5. Compute f(-1) given the function definition:

Answer: 

Quiz 4: Performing function operations

Q1. S​olve for x given: \frac{x}{8}+\frac{1}{3} = \frac{2}{8}8x​+31​=82

Answer:

Q2. S​olve for x, given: 3x = 243x=24

Answer: 

Q3. S​olve for x, given: \sqrt{x+5}=\sqrt{\frac{1}{2}+\frac{1}{8}}x+5​=21​+81​​

Answer: 

Quiz 5: Multiplying Binomials Quiz

Q1. W​hat is (2x-y)^{3}(2x−y)3?

• 8​x3−y3
• 8​x3−6x2y−3xy2−y3
• 8​x3−12x2y−6xy2−y3
• 8​x3−12x2y+6xy2−y3

Quiz 6: Rationalizing Denominators Quiz

Q1. R​ationalize the denominator of \frac{x-3}{\sqrt{x-2}}x−2​x−3

• \frac{(x-3) \cdot \sqrt{x-2}}{x-2}x−2(x−3)⋅x−2
• ​\frac{(x-3) \cdot \sqrt{x-2}}{\sqrt{x-2}}x−2​(x−3)⋅x−2
• \frac{\sqrt{x-3}}{x-2}x−2x−3​

Q2. R​ationalize the denominator of \frac{x+4}{\sqrt{x+6} \sqrt{x+5}}x+6​x+5​x+4

• \frac{x+4 \cdot \sqrt{2x+5+6}}{2x+5+6}2x+5+6x+4⋅2x+5+6​
• \frac{(x+4) \cdot \sqrt{x+6} \cdot \sqrt{x+5}}{x^{2}+11x+30}x2+11x+30(x+4)⋅x+6​⋅x+5
• \frac{(x+4) \cdot \sqrt{x+6} \cdot \sqrt{x+5}}{\sqrt{x+5} \sqrt{x+6}}x+5​x+6​(x+4)⋅x+6​⋅x+5​

Quiz 7: Exponent Rules Quiz

Q1. S​imply \frac{3x^{2}y^{-3}z}{12x^{-1}yz^{3}}12x−1yz33x2y−3z

• 4y4z2x3​
• 4y4z2x−3z
• ​\frac{x^{3}z^{-1}}{4y^{4}z^{1}}
• 4y4z1x3z−1​

Q2. W​hich of the following is equivalent to x^{3}x3

• x2+x
• x⋅x2
• x4−x

Q3. T​he exponential function e^{x}ex exponential function will appear very freqeuntly in your mathematics and data science ventures. The exponent properties we just learned also apply to this function.

G​iven that information, simplify: \frac{e^{2x} \cdot e^{3x} – e^{2x}}{e^{7x}}e7xe2x⋅e3x−e2x

• e4x1​
• e−2x1​−e−5x1
• e−3x1
• e2x1​−e5x1

Quiz 8: Applications of Logarithms

Q1. H​ow long would it take for your money to double if it is invested at 4% compounded continously? (Round to the nearest tenth of a year)

Answer: 

Q2. W​hat percentage rate of return (comounded continously) would be required if you want to double your money in 5 years? (Round to 2 decimal places)

Answer: 

Week 2: Induction Proofs, Limits and Continuity

Quiz 1: Proof by Induction Quiz

Q1. T​o solidfy how a proof by induction works, lets quickly go through a very simple example together.

P​rove that n<n+1n<n+1: (For now just focus on n>=0n>=0)

W​hich of the following could be appropriate base cases to use:

• L​et n=0n=0. Then it follows that 0<10<1 for n<n+1n<n+1
• L​et n=xn=x, where xx is some positive integer. Then x<x+1x<x+1.

Q2. N​ow that we have a base case, which of the following assumptions should we make?

• A​ssume that n<n+1n<n+1
• A​ssume that n<n+1n<n+1

Q3. W​hich of the following inductive steps finalizes our proof:

• W​e can show that n=k+1n=k+1 allows for n<n+1n<n+1 to remain true for any k \geq 0k≥0. (k+1)<(k+1)+1 \rightarrow k+1<k+2 \rightarrow 1<2(k+1)<(k+1)+1→k+1<k+2→1<2 T​hus by induction we see our statement is true for any k \geq 0k≥0
• W​e can show that n=k+1n=k+1 allows for n<n+1n<n+1 to remain true for any k \geq 1k≥1. (k+1)<(k+1)+1 \rightarrow k+1<k+2 \rightarrow 1<2(k+1)<(k+1)+1→k+1<k+2→1<2

Quiz 2: Limits at Infinity Quiz

Q1. W​hat is the lim_{x\to\infty} \frac{x^{1/3}}{x^{1/2}}limx→∞​x1/2x1/3​?

• 0
• L​imit diverges to \infty∞

Q2. W​hat is the lim_{x\to\infty} \frac{5x^{2}}{2x^{2}}limx→∞​2x25x2​?

• 0
• L​imit diverges to \infty∞

Quiz 3: Limits at a Specific Point Quiz

Q1. W​hat are the limits coming from the left and the right of lim_{x\to (-1)}limx→(−1)​ for the following piecewise function?

• limx→(−1)+​=0, lim_{x\to (-1)^-} = 1limx→(−1)−​=1
• limx→(−1)+​=1, lim_{x\to (-1)^-} = 1limx→(−1)−​=1
• limx→(−1)+​=1, lim_{x\to (-1)^-} = 0limx→(−1)−​=0

Quiz 4: Continuity of a Function Quiz

Q1. W​here is the following function continuous?

• E​verywhere
• x=−1
• E​verywhere except x=0x=0
• E​verywhere except x=-1x=−1

Q2. I​s the following function continuous everywhere?

• Yes
• No

Q3. Is the function f(x) = \frac{x-1} {x^2 +x -2}f(x)=x2+x−2x−1​ continuous everywhere? If not, where is it discontinuous?

• Yes
• No

Week 3: Definition of a Derivative

Quiz 1: Derivative Concept Check

Q1. W​hich of the following is the equivalent to f'(x)f′(x)?

• f′(x)=limh→∞​hf(x+h)−f(x)
• f′(x)=limh→0​hf(x+h)−f(x)

Quiz 2: Simple Polynomial Function Quiz

Q1. W​hat is the derivative of f(x)=x^3f(x)=x3?

• 3x2
• x3
• x2

Q2. W​hat is the derivative of f(x)=x^{n-3}f(x)=xn−3 where n is some real number?

• n⋅xn−1
• (n−3)⋅xn−4
• (n−3)⋅xn−2

Quiz 3: Derivative of an Exponential Function Quiz

Q1. W​hat is the derivative of f(x)=5e^{x}f(x)=5ex?

• ex
• 5ex

Quiz 4: Derivative of a Constant Quiz

Q1. W​hat is the derivative of f(x)=x^{3}-3x+4e^{x}f(x)=x3−3x+4ex?

• 3x2−3+4ex−1
• 3x2−3+4ex
• x3−3

Week 4: Product and Chain Rule

Quiz 1: Product Rule Quiz

Q1. W​hat is f'(x)f′(x) if f(x) = (x^3-3x+4)(x^2-1)f(x)=(x3−3x+4)(x2−1)?

• f′(x)=(x3−3x+4)(x2−1)+(3x3−3)(2x)
• f′(x)=(x3−3x+4)(2x)+(3x3−3)(x2−1)
• f′(x)=(3x3−3)(2x)

Q2. S​ay that cc is a constant, then by the product rule, which of the following is true for f'(x)f′(x) when f(x)=c \cdot g(x)f(x)=c⋅g(x)?

• f′(x)=c⋅g′(x)+0⋅g(x)=c⋅g′(x)
• f′(x)=c⋅g′(x)+c⋅g(x)

Quiz 2: Quotient Rule Quiz

Q1. F​ind the derivative of f(x)=\frac{x^2-3x}{x^2+x}f(x)=x2+xx2−3x​:

• f′(x)=(x2+x)2−4x2
• f′(x)=(x2+x)24x2
• f′(x)=(x2+x)23x−5x2

Quiz 3: Chain Rule Quiz

Q1. F​ind the derivative of f(x) = (x^3-2x+5)^4f(x)=(x3−2x+5)4

• 4(x3−2x+5)3⋅(3x2−2x)
• (x3−2x+5)3
• (x3−2x+5)3⋅(3x2−2x)

Q2. F​or a variety of problems, we will often find that we get an equation including the exponential.

Consider the example f(x) = e^{2x^2}f(x)=e2x2. We can view this in the context of the chain rule by formulating it is f(x)=h(g(x))f(x)=h(g(x)), where h(g(x))=e^{2x^2}h(g(x))=e2x2 and g(x) = 2x^2g(x)=2x2. Given that, find the derivative of f(x) = e^{2x^2}f(x)=e2x2.

• f′(x)=e4x
• f′(x)=4xe2x2
• f′(x)=e4x+4⋅(x+1)

Week 5: Using Derivatives to Graph Functions

Quiz 1: Using the Derivative to Graph Functions Quiz

Q1. What does it mean for f'(x)>0f′(x)>0 for some xx?

• The function is decreasing a xx
• T​he function is not changing at xx
• T​he function is increasing at x

Q2. W​here is ff increasing and decreasing for f(x)=x^3-3x^2-9x+4f(x)=x3−3x2−9x+4?

• f is increasing for x<-1x<−1, x>3x>3. ff is decreasing for -1<x<3−1<x<3.
• f is increasing for x<-1x<−1. ff is decreasing for x>-1x>−1.
• f is increasing for x>3x>3. ff is decreasing for x<3x<3.

Q3. C​onsider f(x)=\frac{1}{x}f(x)=x1​, where is the function increasing and decreasing?

• f is always decreasing.
• f is always increasing.
• ff is increasing for x<0x<0. ff is decreasing for x>0x>0.

Quiz 2: Finding Concavity with the Second Derivative Quiz

Q1. W​hat does it mean for f”(x)>0f′′(x)>0?

• f is concave down at xx
• f is concave up at xx

Q2. W​here is ff concave up for f(x)=x^3-3x^2-9x+4f(x)=x3−3x2−9x+4?

• x>6
• x=1
• x<1
• x>1

Quiz 3: Comparing the Graphs of f(x) and f'(x) Quiz

Q1. W​hich of the following derivatives f'(x)f′(x) is graphed below:

• f′(x)=3x2−6x−9
• f′(x)=3x−6
• f′(x)=3x2−6x−12

Q2. W​hich of the following derivatives f'(x)f′(x) is graphed below:

• f′(x)=3x2+3x−9
• f′(x)=3x+3
• f′(x)=3x2+3x−6

Q3. W​hich of the following derivatives f'(x)f′(x) is graphed below?

• f′(x)=x2−4x
• f′(x)=x3−3x2−6x+4
• f′(x)=x3−3x2−9x+4

Q4. W​hich of the following derivatives f'(x)f′(x) is graphed below?

• f′(x)=x3−5x+6
• f′(x)=x3+23​x2−6x+1
• f′(x)=x3+23​x2−5x

Quiz 4: Graphing Functions – A More Complicated Example Quiz

Q1. W​hich of the following derivatives f'(x)f′(x) is graphed below?

• f′(x)=(x2−1)6x
• f′(x)=(x2−1)26x
• f′(x)=x6x2

Q2. W​hich of the following derivatives f'(x)f′(x) is graphed below?

• f′(x)=x2−1x2−4
• f′(x)=x−1x3−4
• f′(x)=x2−4

Week 6: Finding Maximums and Minimums

Quiz: Finding Maximums and Minimums Quiz

Q1. What is the maximum volume you can get for an open box constructed by removing squares of size x from each corner of a paper that is 6 cm by 6 cm and folding up the sides? (Express answer in cm^3cm3)

Answer: 

Q2. B​ased off the graphed function below, is the green point a relative or absolute maximum?

• A​bsolute
• R​elative

Q3. B​ased off the graphed function below, is the green point an absolute or relative minimum?

• A​bsolute
• R​elative

Conclusion:

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