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MAT 210 Final Exam Review
Limits and Continuity
- Calculate the limit: lim
௫→ஶ
ହି଼ ௫
య
ିଽ௫ିସ௫
య
Answer: 2
- Calculate the limit: lim
௫→ஶ
ଷିଶ ௫
మ
ଶିସ௫
మ
ା௫
ఱ
Answer: 0
- Calculate the limit: lim
௫→ଶ
௫
మ
ି ହ௫ ା
௫
మ
ି ଶ௫
Answer: -1/
- Calculate the limit: lim
௫→ଷ
௫
మ
ି ଽ
ଶ௫ି
Answer: 3
- Calculate the limit: lim
௫→ ି
௫
మ
ାଷ௫ିଶ
௫
మ
ା଼௫ା
Answer: 11/
- Calculate the limit: lim
௫→ସ
ష
௫ିଵ
௫ିସ
Answer: −∞
- Calculate the limit: lim
௫→ସ
శ
௫ିଵ
௫ିସ
Answer: ∞
- Let 𝑓(𝑥) = ቄ
Is 𝑓 continuous at 𝑥 = 4? Justify your answer. Answer: No
- Use the graph of function 𝑓(𝑥) answer the questions. Write DNE if a limit does not exist.
(a) lim
௫→
శ
𝑓 (𝑥) Answer: 5
(b) lim
௫→
ష
Answer: 2
(c) lim
௫→
Answer: DNE
(d) 𝑓(6) Answer: 5
(e) lim
௫→ି଼
𝑓 (𝑥) Answer: -
(f) 𝑓(−8) Answer: - 3
(g) lim
௫→ି ଶ
𝑓 (𝑥) Answer: DNE
(h) lim
௫→ଵ
𝑓 (𝑥) Answer: 0
(i) List the 𝑥-values of discontinuous point(s) of the function 𝑓(𝑥). Answer: -8, -2, 6
Rates of Change
- Let 𝑓(𝑥) = 𝑥
ଷ
- Find the average rate of change of 𝑓 over the interval [1, 4].
Answer: 21
- The graph below shows the population of beetles in a greenhouse 𝑡 weeks after the season's flowers were planted.
a) Calculate the average rate of change over the interval [1,10].
Answer: 425/
Answer: 7.8 dollars per box. After 25 boxes of cookies have been sold, the total profit will increase by about
7.8 dollars per additional box sold, or the profit from selling the 26
th
box is about 7.8 dollars.
- Your monthly cost (in dollars) from selling homemade candles is given by
ଶ
, where x is the number of candles you sell in a month. The revenue from selling
𝑥 candles is 𝑅(𝑥) = 7𝑥.
a) Write a function 𝑃(𝑥) for your monthly profit of producing and
selling x candles.
Answer: 𝑃
ଶ
b) Calculate 𝑃( 100 ). Include units.
Answer: 520 dollars
c) Write a function for your marginal profit.
Answer: 𝑀𝑃(𝑥) = 6.9 − 0.004𝑥
d) Calculate your marginal profit if you produce and sell 100 candles. Include units and interpret your answer.
Answer: 6.5 dollars per candle. The profit from selling the 101
st
candle is about 6.5 dollars. Or the total profit
will increase by 6.5 dollars per candle sold, after 100 candles are sold.
Average Velocity and Instantaneous Velocity
- Assume that the distance, s (in meters), traveled by a car moving in a straight line is given by the function
ଶ
− 3𝑡 + 5, where 𝑡 is measured in seconds.
a) Find the average velocity of the car during the time period from 𝑡 = 1 to 𝑡 = 4.
Answer: 2 m/s
b) Find the instantaneous velocity of the car at time 𝑡 = 3 seconds.
Answer: 3 m/s
Chain Rule
- Find the derivative 𝑦′ of each function.
(a) 𝑦 = 0.4( 3 𝑥
ଶ
ହ
(b) 𝑦 = 3 ln(5𝑥
ସ
(c) 𝑦 = 8𝑒
ି ଶ௫
(d) 𝑦 = (𝑥
ଶ
ଶ௫ାଷ
(e) 𝑦 = √𝑥
ଷ
Answer:
(a) 2 (3𝑥
ଶ
ସ
(b)
ଷ(ଶ௫
య
ାସ)
ହ௫
ర
ାସ௫
(c) −16𝑒
ି ଶ௫
(d) (2𝑥 − 2)𝑒
ଶ௫ାଷ
ଶ
ଶ௫ାଷ
ଶ
ଶ௫ାଷ
(e)
ଵ
ଶ
ଷ
ି
భ
మ ∙ (3𝑥
ଶ
Implicit Differentiation
- Find the derivative
ௗ௬
ௗ௫
(a) 𝑥
ଷ
ଷ
(b) 6𝑥
ଶ
ଶ
Answer:
(a)
ௗ௬
ௗ௫
ଷ௫
మ
ଷ௬
మ
ି ଵ
(b)
ௗ௬
ௗ௫
ଵହିଵଶ௬
௫
మ
ି ଶ௬
Absolute and Relative Maxima and Minima
- The function of a function 𝑓 on [−3, 3] is given below.
(a) 𝑓 has a relative minimum at 𝑥 =
(b) 𝑓 has an absolute maximum at 𝑥 =
(c) 𝑓 has an absolute minimum at 𝑥 =
(d) 𝑓′ is zero at 𝑥 =
(e) 𝑓′ is positive on interval(s):
(f) 𝑓′ is negative on interval(s):
Answer:
(a) 0
(b) −1, 1
(c) −3, 3
(d) −1, 0, 1
(e) (−3, −1) and (0, 1)
(f) (−1, 0) and (1, 3)
- Suppose 𝐶
ଶ
- 2𝑥 + 4000 is the total cost for a company to produce 𝑥 units of a certain product. Find
the production level 𝑥 that minimizes the average cost 𝐶
(௫)
௫
Answer: 𝐶
ି ଵ
. Set 𝐶
ᇱ
ି ଶ
= 0 and solve for 𝑥: 𝑥 = 447.
- I need to create a rectangular vegetable patch with an area of exactly 162 square feet. The fencing for the east and
west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. What are the
dimensions of the vegetable patch with the least expensive fence?
Answer:
Minimize the cost 𝐶 = 4𝑥 + 8𝑦 subject to area 𝑥𝑦 = 162, where 𝑥 is the length of the north and south sides, and 𝑦
is the length of east and west sides.
Dimension is 𝑥 = 18, 𝑦 = 9. Minimum cost = 144 dollars.
Higher-order Derivatives, Acceleration and Concavity
- The graph of a function 𝑦 = 𝑓(𝑥) is given below.
(a) 𝑓
ᇱ
ᇱ
(b) Is 𝑓
ᇱᇱ
(−4) is positive or negative? Is 𝑓
ᇱᇱ
( 0 ) is positive or negative?
(c) If 𝑓 has a point of inflection at 𝑥 = −2, then 𝑓
ᇱᇱ
(d) 𝑓 is concave (up/down) on interval (−∞, − 2), and concave (up/down) on interval (−2, ∞).
Answer: (a) 0; (b) negative, positive; (c) 0; (d) down, up.
- Suppose the position of a particle moving on a straight line is 𝑠
ଶ
. Find the particle’s acceleration as a
function of time 𝑡.
Answer: 𝑎
ᇱᇱ
ଵ
ସ
ି
య
మ
Related Rates
- The radius of a circular puddle is growing at a rate of 15 cm/sec. How fast is its area growing at the instant when
the radius is 30 cm?
Answer: 2827 cm
ଶ
/sec
- A rather flimsy spherical balloon is designed to pop at the instant its radius has reached 6 cm. Assuming the
balloon is filled with helium at a rate of 13 cubic centimeters per second, calculate how fast the radius is growing
at the instant it pops.
Answer: 0.03 cm/sec
Elasticity
- The weekly sales of some backpacks is given by 𝑞 = 1080 − 18𝑝, where the 𝑞 represents the quantity of
backpacks sold at price 𝑝.
(a) Find the elasticity of demand at the price of $20. Interpret your answer.
(b) Is the demand at the price $20 elastic, inelastic, or unit elastic? Should the price be raised or lowered from $
to increase the revenue?
(c) What price will maximize the revenue?
(d) What is the maximum weekly revenue?
Answer:
(a) 𝐸(𝑝) = −(−18) ∙
ଵ଼ିଵ଼
ଵ଼
ଵ଼ିଵ଼
, so 𝐸( 20 ) =
ଵ଼ ∙ଶ
ଵ଼ିଵ଼ ∙ଶ
This means the demand will drop by 0.5% for 1% increase from current price $20.
(b) 0.5 < 1, it is inelastic. The price should be raised to increase revenue.
(c) Solve for the price when 𝐸(𝑝) = 1. Solving
ଵ଼
ଵ଼ିଵ଼
= 1 gives 𝑝 = $30.
(d) 𝑅 = 𝑝𝑞 = 30
= 16200 dollars.
- Suppose the demand function is 𝑞 = −2𝑝
ଶ
- 33𝑝, where 𝑞 represents the quantity sold at price 𝑝.
(a) Find the price elasticity of demand 𝐸(𝑝).
(b) Find the elasticity when 𝑝 = $15. If the price increase by 1%, the demand will drop by how much? Should the
price be lowered or raised from $15 to increase the revenue?
Answer:
(a) 𝐸
ିଶ
మ
ାଷଷ
ସିଷଷ
ିଶାଷଷ
(b) 𝐸
ସ(ଵହ)ିଷଷ
ିଶ (ଵହ)ାଷଷ
ଶ
ଷ
= 9 > 1. It is elastic. The demand will drop by 9% if the price increases by 1%.
The price should be lowered from $15 to increase revenue.
Definite Integral; Left Riemann Sum
- Evaluate the definite integrals.
(a) ∫
ହ
ସ
ଶ
ଵ
(b) ∫
ହ
௫
ଶ
(c) ∫
ଵ
௫
మ
ଵ
ଵ
(d) ∫
ି ଷ௫
ଶ
Answer:
(a) 2
(b)
ସହ
ଶ
ଶ
(c)
ଽ
ଵ
(d) 3(1 − 𝑒
ି
- Use a Left Riemann sum to estimate the definite integral with 𝑛 = 4 subintervals.
ଷ
ଶ
Answer: 0.
∆𝑥 = 0.25, LRS = 0.25 ቀ
ଵ
ଵାଶ(ଶ)
ଵ
ଵାଶ(ଶ.ଶହ)
ଵ
ଵାଶ(ଶ.ହ)
ଵ
ଵାଶ(ଶ.ହ)
Fundamental Theorem of Calculus; Applications of Definite Integrals
- A particle moves in a straight line with velocity 𝑣
ଶ
- 8 meters per second, where 𝑡 is time in seconds.
Find the displacement of the particle between 𝑡 = 2 and 𝑡 = 6 seconds.
Answer: −37 meters
Displacement = 𝑠( 6 ) − 𝑠( 2 ) = ∫ 𝑣(𝑡)
ଶ
ଶ
ଶ
ଵଵଶ
ଷ
≈ −37 meters.
- The marginal revenue of the 𝑥th box of flash cards sold is 500 𝑒
ି .ଵ௫
dollars. Find the revenue generated by
selling box 101 through 5,000.
Answer: 448,598 dollars
Total revenue generated = 𝑅( 5000 ) − 𝑅( 101 ) = ∫
ହ
ଵଵ
ି .ଵ௫
ହ
ଵଵ
𝑑𝑥 ≈ 448597.54 dollars
- Since YouTube first became available to the public in mid-2005, the rate at which video has been uploaded to this
site can be approximated by 𝑓
ଶ
− 2.6𝑡 + 2.3 million hours of videos per year (0 ≤ 𝑡 ≤ 9), where 𝑡 is
time in years since June 2005. Use a definite integral to estimate the total number of hours of video uploaded from
June 2007 to June 2010.
Answer: 23 million hours of video
Total number of hours = ∫ 𝑓
ହ
ଶ
ଶ
ହ
ଶ
𝑑𝑡 ≈ 23 million hours of video
- Calculate the area of the region bounded by 𝑦 = √
𝑥, the 𝑥-axis, and the lines 𝑥 = 0 and 𝑥 = 16.
Answer:
ଵଶ଼
ଷ
Area under curve = ∫ √
ଵ
ଵଶ଼
ଷ
Integration by Parts: ∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢
- Use integration by parts to find the integrals.
(a) ∫
ି ହ௫
(b) ∫ 𝑥
ଶ
ln 𝑥 𝑑𝑥
Answer:
(a) −
ଵ
ହ
ି ହ௫
ଷ
ଶହ
ି ହ௫
ଷ
ହ
ଶଷ
ଶହ
ି ହ௫
(Let 𝑢 = 3𝑥 + 4, 𝑑𝑣 = 𝑒
ି ହ௫
𝑑𝑥. Then 𝑑𝑢 = 3𝑑𝑥 and 𝑣 = −
ଵ
ହ
௫
(b)
ଵ
ଷ
ଷ
ln 𝑥 −
ଵ
ଽ
ଷ
ଶ
Area between Curves
- Find the area of the region enclosed by the curves of 𝑓(𝑥) = 𝑥
ଶ
− 𝑥 + 5 and 𝑔(𝑥) = 𝑥 + 8.
Answer:
ଷଶ
ଷ
Area = ∫
ଶ
ଶ
ିଵ
ଶ
ଶ
ିଵ
௫
య
ଷ
ଶ
ି ଵ
ଶ
32
3
Improper Integral
- Determine whether each improper integral is convergent or divergent. If it is convergent, find its value.
(a) ∫
଼
௫
మ
ஶ
ଵ
(b) ∫
ି ଶ௫
ஶ
ସ
(c) ∫
ଵ
௫
ஶ
ଵ
(d) ∫ 𝑥
ଶ
ஶ
ଷ
(e) ∫
ଶ௫
ିஶ
Answer:
(a) Converges to 8
ଶ
ஶ
ଵ
𝑑𝑥 = lim
௧→ஶ
ି ଶ
௧
ଵ
𝑑𝑥 = lim
௧→ஶ
ଵ
௧
൰ = lim
௧→ஶ
(b) Converges to
ଵ
ଶ
ఴ
ି ଶ௫
ஶ
ସ
= lim
௧→ஶ
ି ଶ௫
௧
ସ
𝑑𝑥 = lim
௧→ஶ
ି ଶ௫
ସ
௧
൰ = lim
௧→ஶ
ି ଶ௧
ି ଶ (ସ)
ି଼
଼
(c) Diverges
ஶ
ଵ
𝑑𝑥 = lim
௧→ஶ
௧
ଵ
𝑑𝑥 = lim
௧→ஶ
(ln 𝑡 − ln 1) = ∞ − 1 = ∞
(d) Diverges
(e) Converges to
ଵ
ଶ
