Lab08 Concepts Review.ipynb

Kernel: SageMath 8.3

Lab08 Concepts Review

Replace with group member names

Peyton Smith Brayton Yoder

Learning Objectives

1. State formal definitions of function, derivative, and definite integral.

2. Find derivatives and definite integrals for functions described verbally, numerically, graphically, and symbolically.

3. Describe the concepts of function, derivative, and definite integral to someone wanting a brief introduction to these concepts.

Procedure

1. Work in groups of one to three students. Collaboration is encouraged. Take turns at the keyboard while working jointly, or discuss discoveries and questions while working separately. A group of up to three students can submit a single report.

2. Complete the exercises.

3. If this notebook is the one to be graded for your group, make sure all group member names appear in the first cell and rename the file to "GRADE ME Lab08 Concepts Review.ipynb".

Definitions (9 points)

State formal definitions for function, derivative, and definite integral.

$$y=f(x)=x$$ is the formal defentition of a function

$$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.$$ is the formal defenition of a derivative

$$\int_a^b f(x)\,dx$$ is the formal defnition of a defnitie integral

Verbal (15 points)

A yam has just been taken out of the oven and is cooling off before being eaten. The temperature, (T), of the yam (measured in degrees Fahrenheit) is a function of how long it has been out of the oven, (t) (measured in minutes). Thus, we have (T=f(t)).

1. (3 points) What are the units and sign of (f'(t))? Why?

2. (3 points) What are the units and sign of (f\text{''}(t))? Why?

3. (3 points) What are the units and sign of (\int_0^{30} f(t) , dt)? Why?

After (t) hours, the velocity of a particle moving along the (x)-axis is (f(t)) miles per hour.

4. (3 points) Interpret and give the units for (f'(2.3)).

5. (3 points) Interpret and give the units for (\int_1^2 f(t) , dt).

1). The units would be farhenheit/minute because and the sign is negtive because as the yam is out of the oven longer it gets colder

2). the unites are farhenheit/min^2 and the sign is negative because the rate of change decreases as time goes on

3). The units are degrees farhenheit and the sign is negtive because the net change in temperature for the yams decreses over 30 minutes

4). The rate of change 2.3 hours after the particle started moving

5). MPH, the change in speed between 1 and 2 hours

Numerical (18 points)

Suppose the known values of a decreasing function (f) are given in the following table.

($$\begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 3 & 6 & 8 & 9 \\ \hline \text{f(x)} & 10 & 8 & 6 & 4 & 2 \\ \hline \end{array}$$)

1. (3 points) Estimate (f'(6)). Show how you came up with your estimate.

2. (3 points) Estimate (f\text{''}(6)). Show how you came up with your estimate.

3. (3 points) Estimate $\int_1^9 f(t) \, dt$ by calculating a lower bound, an upper bound, and a best estimate.

The table shows (f(t)), total sales of music compact discs (CDs), in millions, as a function of year (t).

($$\begin{array}{|c|c|c|c|c|c|} \hline \text{Year}, t & 1994 & 1996 & 1998 & 2000 & 2002 \\ \hline \text{CD} \text{sales}, f(t) & 662.1 & 778.9 & 847.0 & 942.5 & 803.3 \\ \hline \end{array}$$)

4. (3 points) Estimate (f'(2002)). Give units with and interpret your answer.

5. (3 points) Use (f'(2002)) to estimate (f(2010)). Give units with and interpret your answer.

6. (3 points) Estimate $\frac{1}{6}\int_{1996}^{2002} f(t) \, dt$. Give units with and interpret your answer.

1). f'(6)=-1

4-6/8-6= -1

2).

3).

4). -69.9 million CDs sold per year. This means that everyear 69.9 million less CDs are sold.

5). 176.6 Million cds will be sold in 2010

Graphical (18 points)

Suppose the function (f) is defined by the graph below.

1. (3 points) Estimate (f'(3)).

2. (3 points) On what interval(s) is (f(x)>0)?

3. (3 points) On what interval(s) is (f'(x)>0)?

4. (3 points) On what interval(s) is (f\text{''}(x)>0)?

5. (6 points) Estimate $\int_0^5 f(x) \, dx$ using a lower sum, an upper sum, and an average of the lower and upper sums.

n((-16)-11)/(4-2)

-13.5000000000000

1. f'(3)= -13.5

2. f(x)>0 on the interval from 0 to 3

3. f'(x)>0 on the interval from 0 to 1.8, and the interval from 4.2 to 5

4. f''(x)>0 on the interval from 1.8 to 5

Symbolic (15 points)

Let (g(x)=\ln \left(x^3+4\right)e^{5\sin (x)}).

1. (6 points) Find (g'(x)) using the derivative rules. Show your work step by step and state which rule or rules are used in each step. (You may do these exercises on paper, and then scan a picture of your work to this folder.)

2. (3 points) Find the exact value of (g'(0)) in a simplified form.

3. (3 points) Use plot(...) to plot (g), (g'), and (g\text{''}) on a single pair of axes on the domain ([-1,2\pi ]). Clearly indicate (possibly with a legend) to which function each curve belongs.

4. (3 points) Estimate $\int_{-1}^{2\pi } g(x) \, dx$ to the nearest integer.

Informal Summary (20 points)

Describe what a function is. State four different ways of representing a function. Describe derivative and definite integral for the four ways of representing a function. The purpose of this part of the lab is to deepen your own understanding by writing descriptions that would be understood by a student just starting this course. So, it is okay to use simple examples and informal language.

A function is a rule that for every unique input there is a corresponding output. Function can be shown by words, graphs, or even verbally

Acknowledgments

Acknowledge all people outside of the group who provided assistance and any paper or internet sources used to help complete the report. Citations should be sufficient to allow a reader to find the sources. If you do not use any outside assistance or resources, make a statement of that in this section.

Brayton Yoder

Final Check (5 points)

Make sure all group member names appear in the first cell, all exercises are completed, no extraneous cells remain, the acknowledgments section has been completed, and the file has been renamed to "GRADE ME Lab08 Concept Review.ipynb".