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MATH120: ELEMENTARY CALCULUS NOTES AND EXAMPLES FROM CLASS...

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Name: _______________________ MATH 120

TAs name: _________________ Group Worksheet

Section number: _______ 4.1, 4.2, 4.3, 4.4

READ AND FOLLOW DIRECTIONS CAREFULLY: x Youll ork on the questions as a group, but ill each hand in our on orksheet. Use the back of the page if necessar. Your TA is a resource, but will not be doing the work for you. x These worksheets will be graded for each discussion out of a total possible of 10 points, and ill constitute the discussion qui part of our final average. x We plan on 14 of these worksheets, and will be dropping the lowest 4 scores at the end of the semester. Thus, discussions missed because of illness, court, athletics, holydays, oversleeping, etc., etc. will not count against you. x Show enough work that we can follow your thinking. You must show all appropriate calculus work in order to receive full credit for an answer.

1. Simplify each of

2

2∗2

1 𝑥

, 25𝑥/4 ∗ 󰇡 󰇢 , 2

3

18

󰇛2 𝑥+1 ∗ 2−3 󰇜2  2 and 2 𝑥 

2. Solve the equations for x

8

2

0

 𝑥 𝑒 𝑥 3. Differentiate 𝑦  8𝑒 𝑥 󰇛1  2𝑒 𝑥 󰇜2 and √

4. Differentiate 𝑦  𝑒 √𝑥

 +1

and 𝑦  𝑒 3 󰇛𝑒 2  𝑒 4 󰇜

1

5. Simplify 𝑒 ln 3−2 ln 5 and ln 󰇡  󰇢 𝑒

4  ln 𝑥  0 , 𝑙𝑛󰇛𝑥 2  5󰇜  0 , 𝑙𝑛󰇛ln 3𝑥󰇜  0

6. Solve the equations for x





7. Differentiate 2 𝑥 and 25𝑥 Hint: Express 2 𝑥 and 25𝑥 of the form 𝑒 󰇛𝑥󰇜 using the properties of natural log and the exponential function

Name: _______________________ MATH 120

TAs name: _________________ Group Worksheet

Section number: _______ 4.5, 5.1, 5.2, 5.4

READ AND FOLLOW DIRECTIONS CAREFULLY: x Youll ork on the questions as a group, but ill each hand in our on orksheet. Use the back of the page if necessar. Your TA is a resource, but will not be doing the work for you. x These worksheets will be graded for each discussion out of a total possible of 10 points, and ill constitute the discussion qui part of our final average. x We plan on 14 of these worksheets, and will be dropping the lowest 4 scores at the end of the semester. Thus, discussions missed because of illness, court, athletics, holydays, oversleeping, etc., etc. will not count against you. x Show enough work that we can follow your thinking. You must show all appropriate calculus work in order to receive full credit for an answer.

1. Write the equation of the tangent line to the graph of 𝑦  ln󰇛𝑥 + 𝑒󰇜 at 𝑥  1

ln󰇛𝑥󰇜+1

2. The function 𝑓󰇛𝑥󰇜  𝑥 relative maximum point?

has a relative extreme point for 𝑥  0. Find the coordinates of the point. Is it a

3. Solve the differential equation 𝑃󰆒 󰇛𝑡󰇜  −0.6𝑃󰇛𝑡󰇜, 𝑃󰇛0󰇜  50.

4. After t hours there are P(t) cells present in a culture, where 𝑃󰇛𝑡󰇜  300𝑒 0.01𝑡 a) How many cells were present initially? b) Give a differential equation satisfied by P(t) c) At what time will the initial population double d) At what time will the population equal 1500? e) What is the rate of increasing when the population equals 1500?

5. Ten grams of a radioactive material disintegrates to 2 grams in 5 years. a) What is the decay constant? b) What is the half-life of the radioactive material? c) How long it will take until the material disintegrates to 1 gram

6. $1000 is deposited in a savings account at 6% yearly interest compounded continuously. a) How many years are required for the balance in the account to reach $2500 b) How much money are there in the account after 2 years

7. A news item is spread by word of mouth to a potential audience of 10,000 people. After t days 10,000 𝑓󰇛𝑡󰇜  1+50𝑒 −0.𝑡 people will have heard the news. a) Show that the function is increasing for 𝑡  0 b) What is the maximum number of people that will know the news?

Name: _____ _________________ MATH 120

Section number: _______

TA ame: _________________ Group Worksheet

6.1

You must show all appropriate calculus work in order to receive full credit for an answer. 1. Find all antiderivative of each of the following functions: a) 𝑓󰇛𝑥 󰇜  9𝑥 8 b) 𝑓󰇛𝑥 󰇜  𝑒 −3𝑥 c) 𝑓󰇛𝑥 󰇜  3 d) 𝑓󰇛𝑥 󰇜  4𝑥

2. Find each of the following integrals: a) ∫ 7𝑑𝑥

𝑥

b) ∫ 3 𝑑𝑥

2

𝑥

c) ∫ 󰇡 𝑥 + 2 󰇢 𝑑𝑥

d) ∫ 󰇡

2 √𝑥

+ 2√𝑥󰇢 𝑑𝑥

e)∫ 󰇡3𝑒 −𝑥 + 2𝑥 

𝑒 0.𝑥 󰇢 𝑑𝑥 2

3. Find all functions f(x) that satisfy the given conditions: a) 𝑓 󰆒 󰇛𝑥 󰇜  2𝑥  𝑒 −𝑥 , 𝑓󰇛0󰇜  1 b) 𝑓 󰆒 󰇛𝑥 󰇜  𝑥 2 + √𝑥, 𝑓󰇛1󰇜  3

4. A ball is thrown upward from a height of 256 feet above the ground with an initial velocity of 96 feet per second. From physics it is known that the velocity at time t is 𝑣󰇛𝑡󰇜  96  32𝑡 feet per second. a) Find s(t), the function giving the height above the ground of the ball at time t b) How long will the ball take to reach the ground? c) How high will the ball go?

5. A small tie shop finds that at a sales level of x ties per day, its marginal profit is MP(x) dollars per tie, where 𝑀𝑃󰇛𝑥󰇜  1.30 + 0.6𝑥  0.0018𝑥 2 . Also, the shop will lose $95 per day at a sales level of 𝑥  0. Find the profit from operating the shop at a sales level of x ties per day.

Name: _________ _____________ TA ame: _________________ Section number: _______ MATH 120 Group Worksheet 6.2, 6.3 You must show all appropriate calculus work in order to receive full credit for an answer. 1. Evaluate the given integrals 4 𝑥 −√𝑥

a) ∫1

𝑥

𝑑𝑥

−1 1+𝑥

b) ∫−2

𝑑𝑥

𝑥

1   +.  

c) ∫0

10

10

1

2. Given ∫−1 𝑓󰇛𝑥󰇜𝑑𝑥  1 and ∫−1 𝑓󰇛𝑥󰇜𝑑𝑥  4 , find ∫1

𝑑𝑥

𝑑𝑥

𝑓󰇛𝑥󰇜𝑑𝑥

3

3

 2   + 2

d) ∫0

3

3. Given ∫−0.5 𝑓󰇛𝑥󰇜𝑑𝑥  1 and ∫−0.5 2𝑔󰇛𝑥󰇜  3𝑓󰇛𝑥󰇜𝑑𝑥  4 , find ∫−0.5 𝑔󰇛𝑥󰇜𝑑𝑥

4. Given 𝑓 󰆒 󰇛𝑡󰇜  12𝑡 



, compute 𝑓󰇛3󰇜  𝑓󰇛0󰇜

5. The velocity at time t seconds of a ball thrown up into the air is 𝑣󰇛𝑡󰇜  32𝑡  75 feet per second. a) Compute the displacement of the ball during the time interval 1  𝑡  3 b) Compute the displacement of the ball during the time interval 1  𝑡  5 c) Explain why the second displacement is smaller

6. Find the area under each of the given curves a) 𝑦  √𝑥, 𝑥  0 𝑡𝑜 𝑥  4 b) 𝑦  󰇛2𝑥  3󰇜4 , 𝑥  1 𝑡𝑜 𝑥  4

7. Use a Riemann sum to approximate the area under the graph of 𝑓󰇛𝑥󰇜  𝑥2 on 2  𝑥  2 using 4 subintervals and the midpoints of the subintervals.

Name: _______________________ MATH 120

TA ame: _________________ Group Worksheet

Section number: _______ 6.4, 6.5

You must show all appropriate calculus work in order to receive full credit for an answer. 1. Find the area of the region between the curve 𝑓󰇛𝑥󰇜  𝑒 −𝑥  2 and the x-axis from -1 to 2

2. Find the area of the region between the curves 𝑦  𝑥󰇛2  𝑥󰇜 and 𝑦  2 from x=0 to x=2

4

3. Find the area of the region bounded by the curves 𝑦  𝑥 and 𝑦  5  𝑥

1

4. Find the area of the region bounded by 𝑦  𝑥 , 𝑦  4𝑥, 𝑦 

𝑥 2

for 𝑥  0

5. During a certain 12-hour period, the temperature at time t (measured from the start of the period) was 1 𝑇󰇛𝑡󰇜  47  4𝑡  𝑡 2 degrees. What was the average temperature during that period? 3

6. Find the consmers srpls for the demand curve 𝑝󰇛𝑥󰇜  √16  0.02𝑥 at the given sales level x=350

7. At a certain supermarket, the amount of wait time at the express lane is a random variable with probability 11 density function 𝑓󰇛𝑥󰇜  2 for 0  𝑥  10 minutes. Find the probability of having to wait less that 4 10󰇛𝑥+1󰇜

minutes at the express lane.

8. Let Z be a standard normal random variable. Calculate a) Pr󰇛1.3  𝑍  0󰇜 b) Pr󰇛𝑍  0.25󰇜 c) Pr󰇛Z  1.34󰇜

Name: _______________________ TA ame: _________________ Section number: _______ MATH 120 Group Worksheet 7.1-7.2 You must show all appropriate calculus work in order to receive full credit for an answer. 1. Let 𝑔󰇛𝑥, 𝑦󰇜  𝑥2 + 2𝑦 3 and ℎ󰇛𝑥, 𝑦, 𝑧󰇜  𝑥2 𝑒 𝑦

2 +𝑧 2

. Compute 𝑔󰇛3, 2󰇜 and ℎ󰇛2, 3, 4󰇜

2. Find a formula C(x, y, z) that gives the cost of materials for an open rectangular box (x=length, y=width, z=height). Assume that the materials for the bottom cost $3 per square foot and for the sides cost $5.

3. The value of residential property for tax purposes is usually much lower than the actual market value. If v is the market value, the assessed value for real estate taxes may be only 40% of v. Supposed that the property tax, T, 𝑟 󰇛0.40𝑣  𝑥󰇜 where v is the market value of a in a community is given by the function 𝑇  𝑓󰇛𝑟, 𝑣, 𝑥󰇜  00 e  da,   a ee ee (a be f da deedg  e e f e), ad r is the tax rate (stated in dollars per hundred dollars. Determine the real estate tax on a property valued at $100,000  a ee ee f $5000, ag a a ae f $2.20 e ded da f e assessed value.

4. Given the function 𝑧  2𝑥  3𝑦 , find the equation for the level curve for which z=6. Write your answer as y=...

5. Find both partial derivatives 𝑥

𝑦

a) 𝑓󰇛𝑥, 𝑦󰇜  𝑦 + 𝑥

𝑓 𝑥

𝑓

and 𝑦 for each of the following functions

b) 𝑓󰇛𝑥, 𝑦󰇜 

6. Let 𝑓󰇛𝑥, 𝑦󰇜  𝑥3 𝑦 + 2𝑥𝑦 2 Find

𝑥−𝑦

𝑥+𝑦

c)𝑓󰇛𝑥, 𝑦󰇜  𝑥𝑒 𝑥

2𝑓 , 𝑥 2

 2𝑓 , 𝑦2

 2𝑓 , 𝑥𝑦

2𝑦 2

 2𝑓 𝑦𝑥

7. The productivity of a country is given by 𝑓󰇛𝑥, 𝑦󰇜  300𝑥2/3 𝑦 /3, where x and y are the amount of labor and capital respectively. a) Compute the marginal productivity of labor and capital when x=125 and y=64 b) Use part (a) to determine the approximate effect on productivity of increasing the capital from 64 to 66 units, while keeping labor fixed at 125 units c) What would be the approximate effect of decreasing labor from 125 to 124 units while keeping capital fixed at 64 units?...