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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.

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*+2*x*−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*)=4*e*−2*t* 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: `

Q1. What is the domain and range of the following function?

- Domain: all real numbers, Range: all real numbers
- Domain: x>=-1, Range: all real numbers
- Domain: all real numbers, Range: y>=-1
- Domain: x>=-1, Range: y>=-1

Q2. What is the domain and range of the following function?

- Domain: all Real numbers, Range: all Real numbers
- Domain: x=2, Range: all Real numbers
- Domain: all Real numbers, Range: y=2
- Domain: x=2, Range: y=2

Q3. What is the domain and range of the following function?

- Domain: all Real numbers, Range: all Real numbers
- Domain: x<=3, Range: y>=-1
- Domain: all Real numbers, Range: y>=-1
- Domain: x<=3, Range: all Real numbers

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

`Answer: `

Q2. Using figure A., what is f(-5)?

`Answer: `

Q3. Using Figure A., what is f(1)?

`Answer: `

Q4. Using Figure A., what is f(2)?

`Answer: `

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

`Answer: `

Q1. Solve for x given: \frac{x}{8}+\frac{1}{3} = \frac{2}{8}8*x*+31=82

`Answer:`

Q2. Solve for x, given: 3x = 243*x*=24

`Answer: `

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

`Answer: `

Q1. What is (2x-y)^{3}(2*x*−*y*)3?

- 8
*x*3−*y*3 - 8
*x*3−6*x*2*y*−3*xy*2−*y*3 - 8
*x*3−12*x*2*y*−6*xy*2−*y*3 - 8
*x*3−12*x*2*y*+6*xy*2−*y*3

Q1. Rationalize 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*−2*x*−3

Q2. Rationalize 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}2
*x*+5+6*x*+4⋅2*x*+5+6 - \frac{(x+4) \cdot \sqrt{x+6} \cdot \sqrt{x+5}}{x^{2}+11x+30}
*x*2+11*x*+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

Q1. Simply \frac{3x^{2}y^{-3}z}{12x^{-1}yz^{3}}12*x*−1*y**z*33*x*2*y*−3*z*

- 4
*y*4*z*2*x*3 - 4
*y*4*z*2*x*−3*z* - \frac{x^{3}z^{-1}}{4y^{4}z^{1}}
- 4
*y*4*z*1*x*3*z*−1

Q2. Which of the following is equivalent to x^{3}*x*3

*x*2+*x**x*⋅*x*2*x*4−*x*

Q3. The 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.

Given that information, simplify: \frac{e^{2x} \cdot e^{3x} – e^{2x}}{e^{7x}}*e*7*x**e*2*x*⋅*e*3*x*−*e*2*x*

*e*4*x*1*e*−2*x*1−*e*−5*x*1*e*−3*x*1*e*2*x*1−*e*5*x*1

Q1. How 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. What 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: `

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

Prove that n<n+1*n*<*n*+1: (For now just focus on n>=0*n*>=0)

Which of the following could be appropriate base cases to use:

- Let n=0
*n*=0. Then it follows that 0<10<1 for n<n+1*n*<*n*+1 - Let n=x
*n*=*x*, where x*x*is some positive integer. Then x<x+1*x*<*x*+1.

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

- Assume that n<n+1
*n*<*n*+1 - Assume that n<n+1
*n*<*n*+1

Q3. Which of the following inductive steps finalizes our proof:

- We can show that n=k+1
*n*=*k*+1 allows for n<n+1*n*<*n*+1 to remain true for any k \geq 0*k*≥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 Thus by induction we see our statement is true for any k \geq 0*k*≥0 - We can show that n=k+1
*n*=*k*+1 allows for n<n+1*n*<*n*+1 to remain true for any k \geq 1*k*≥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

Q1. What is the lim_{x\to\infty} \frac{x^{1/3}}{x^{1/2}}*limx*→∞*x*1/2*x*1/3?

- 0
- Limit diverges to \infty∞

Q2. What is the lim_{x\to\infty} \frac{5x^{2}}{2x^{2}}*limx*→∞2*x*25*x*2?

- 0
- Limit diverges to \infty∞

Q1. What 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)^-} = 1*limx*→(−1)−=1*limx*→(−1)+=1, lim_{x\to (-1)^-} = 1*limx*→(−1)−=1*limx*→(−1)+=1, lim_{x\to (-1)^-} = 0*limx*→(−1)−=0

Q1. Where is the following function continuous?

- Everywhere
*x*=−1- Everywhere except x=0
*x*=0 - Everywhere except x=-1
*x*=−1

Q2. Is the following function continuous everywhere?

- Yes
- No

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

- Yes
- No

Q1. Which of the following is the equivalent to f'(x)*f*′(*x*)?

*f*′(*x*)=lim*h*→∞*hf*(*x*+*h*)−*f*(*x*)*f*′(*x*)=lim*h*→0*hf*(*x*+*h*)−*f*(*x*)

Q1. What is the derivative of f(x)=x^3*f*(*x*)=*x*3?

- 3
*x*2 *x*3*x*2

Q2. What 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

Q1. What is the derivative of f(x)=5e^{x}*f*(*x*)=5*ex*?

*ex*- 5
*ex*

Q1. What is the derivative of f(x)=x^{3}-3x+4e^{x}*f*(*x*)=*x*3−3*x*+4*ex*?

- 3
*x*2−3+4*ex*−1 - 3
*x*2−3+4*ex* *x*3−3

Q1. What is f'(x)*f*′(*x*) if f(x) = (x^3-3x+4)(x^2-1)*f*(*x*)=(*x*3−3*x*+4)(*x*2−1)?

*f*′(*x*)=(*x*3−3*x*+4)(*x*2−1)+(3*x*3−3)(2*x*)*f*′(*x*)=(*x*3−3*x*+4)(2*x*)+(3*x*3−3)(*x*2−1)*f*′(*x*)=(3*x*3−3)(2*x*)

Q2. Say that c*c* 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*)

Q1. Find the derivative of f(x)=\frac{x^2-3x}{x^2+x}*f*(*x*)=*x*2+*xx*2−3*x*:

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

Q1. Find the derivative of f(x) = (x^3-2x+5)^4*f*(*x*)=(*x*3−2*x*+5)4

- 4(
*x*3−2*x*+5)3⋅(3*x*2−2*x*) - (
*x*3−2*x*+5)3 - (
*x*3−2*x*+5)3⋅(3*x*2−2*x*)

Q2. For 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*)=*e*2*x*2. 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*))=*e*2*x*2 and g(x) = 2x^2*g*(*x*)=2*x*2. Given that, find the derivative of f(x) = e^{2x^2}*f*(*x*)=*e*2*x*2.

*f*′(*x*)=*e*4*x**f*′(*x*)=4*xe*2*x*2*f*′(*x*)=*e*4*x*+4⋅(*x*+1)

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

- The function is decreasing a x
*x* - The function is not changing at x
*x* - The function is increasing at x

Q2. Where is f*f* increasing and decreasing for f(x)=x^3-3x^2-9x+4*f*(*x*)=*x*3−3*x*2−9*x*+4?

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

Q3. Consider f(x)=\frac{1}{x}*f*(*x*)=*x*1, where is the function increasing and decreasing?

*f*is always decreasing.*f*is always increasing.- f
*f*is increasing for x<0*x*<0. f*f*is decreasing for x>0*x*>0.

Q1. What does it mean for f”(x)>0*f*′′(*x*)>0?

*f*is concave down at x*x**f*is concave up at x*x*

Q2. Where is f*f* concave up for f(x)=x^3-3x^2-9x+4*f*(*x*)=*x*3−3*x*2−9*x*+4?

*x*>6*x*=1*x*<1*x*>1

Q1. Which of the following derivatives f'(x)*f*′(*x*) is graphed below:

*f*′(*x*)=3*x*2−6*x*−9*f*′(*x*)=3*x*−6*f*′(*x*)=3*x*2−6*x*−12

Q2. Which of the following derivatives f'(x)*f*′(*x*) is graphed below:

*f*′(*x*)=3*x*2+3*x*−9*f*′(*x*)=3*x*+3*f*′(*x*)=3*x*2+3*x*−6

Q3. Which of the following derivatives f'(x)*f*′(*x*) is graphed below?

*f*′(*x*)=*x*2−4*x**f*′(*x*)=*x*3−3*x*2−6*x*+4*f*′(*x*)=*x*3−3*x*2−9*x*+4

Q4. Which of the following derivatives f'(x)*f*′(*x*) is graphed below?

*f*′(*x*)=*x*3−5*x*+6*f*′(*x*)=*x*3+23*x*2−6*x*+1*f*′(*x*)=*x*3+23*x*2−5*x*

Q1. Which of the following derivatives f'(x)*f*′(*x*) is graphed below?

*f*′(*x*)=(*x*2−1)6*x**f*′(*x*)=(*x*2−1)26*x**f*′(*x*)=*x*6*x*2

Q2. Which of the following derivatives f'(x)*f*′(*x*) is graphed below?

*f*′(*x*)=*x*2−1*x*2−4*f*′(*x*)=*x*−1*x*3−4*f*′(*x*)=*x*2−4

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^3*cm*3)

`Answer: `

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

- Absolute
- Relative

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

- Absolute
- Relative

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