Calculus

Calculus

2.04
#6. (Round your answers to five decimal places)
For the limit below, find values of δ that correspond to the ε values.

ε δ
0.05
0.01

#7. (Round your answers to five decimal places)

ε δ
0. 5
0.05

#8. Given that
lim
x→4 (4x − 9) = 7,
illustrate Definition 2 by finding values of δ that correspond to
ε = 0.1

δ ≤
______
ε = 0.05
δ ≤
______

and ε = 0.01
δ ≤

______

Section 3.08
#6. If $600 is borrowed at 9% interest, find the amounts due at the end of 4 years if the interest is compounded as follows. Round to the nearest cent.

Weekly ______ dollars
Daily_______ Dollars
Hourly _______ dollars
Continuously ______ dollars

#8. A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. (Round your answer to the nearest whole number.)
(a) If the temperature of the turkey is 150°F after half an hour, what is the temperature after 60 minutes?
T(60) =126 °F (b) When will the turkey have cooled to 110°?
t＝____ mins

#9. A sample of a radioactive substance decayed to 95% of its original amount after a year. (Round your answers to two decimal places.)

(a) What is the half-life of the substance?
_______ yr

(b) How long would it take the sample to decay to 55% of its original amount?
_______ yr
Section 3.09

#2. The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the diameter is 40 mm? Evaluate your answer numerically. (Round the answer to the nearest whole number.)

_______ mm3/s

#11. Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor h = 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 3.5 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q? (Round your answer to two decimal places.)

______ ft/s
#15. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4 = C, where C is a constant. Suppose that at a certain instant the volume is 850 cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what rate is the volume increasing at this instant? (Round your answer to the nearest whole number.)

______ cm3/min

#17. Two sides of a triangle have lengths 11 m and 18 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°? (Round your answer to three decimal places.)
______ m/min
#19. Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 13 cm to 23 cm at a constant rate, how fast was this species’ brain growing when the average length was 17 cm? (Round your answer to four significant figures.)

___________ g/yr

Section 3.10
The circumference of a sphere was measured to be 72 cm with a possible error of 0.5 cm.

What is the relative error? (Round your answer to three decimal places.)
________

#4. Compute Δy and dy for x = 4 and dx = Δx = 1. (Round the answers to three decimal places.)
y = √x
dy =____
Δy =_____
#5. Use a graphing calculator or computer to verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.01. (Round the answers to three decimal places.)

(______, ________)

#8. Compute Δy and dy for x = 5 and dx = Δx = -0.2. (Round the answers to three decimal places.)
y = 5x – x2
dy =____
Δy =_____

#12. Compute Δy and dy for x = 0 and dx = Δx = 0.1. (Round the answers to three decimal places.)
y = ex
dy =____
Δy =_____

#13. The radius of a circular disk is given as 23 cm with a maximum error in measurement of 0.2 cm.
(a) Use differentials to estimate the maximum error in the calculated area of the disk. (Round your answer to two decimal places.)
_____ cm2
(b) What is the relative error? (Round your answer to four decimal places.)
________
©What is the percentage error? (Round your answer to two decimal places.)
______ %

#14, Find the linearization L(x) of the function at a.
f(x) = x2/3, a = 8
L(x) =_________
#15. Use a linear approximation (or differentials) to estimate the given number. (Use the linearization of 1/x. Do not round your answer.)
1/1003
_____

#17.
EXAMPLE 4 The radius of a sphere was measured and found to be 17 cm with a possible error in measurement of at most 0.03 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?
SOLUTION If the radius of the sphere is r then its volume is V = (4/3)πr3. If the error in the measured value of V is ΔV, which can be approximated by the differential
dV=______ πr_______dr
When r = 17 and dr = 0.03, this becomes

dV = 4π( _____)2_____ _____ (Round to the nearest integer.)
The maximum error in the calculated volume is about_____
#18. Use a graphing calculator or computer to verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Round the answers to three decimal places.
Section 4.1
#6. Find the critical numbers of the function on the interval 0 ≤ θ < 2π.

f(θ) = 2cos(θ) + (sin(θ))2
θ =_____ (smaller value)
θ =______(larger value)
#14. Find the absolute maximum and absolute minimum values of f on the given interval. (Round all answers to two decimal places.)
f(t) = t + cot(t/2)

______ (min)
______(max)
#15. Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = e-x – e-2x
[0, 1]
______ (min)
______(max)
#16. If a and b are positive numbers, find the maximum value of f(x) = xa(9 − x)b on the interval 0 ≤ x ≤ 9.

_______

Section 4.03

#3. Find the local maximum value of f using both the First and Second Derivative Tests.
f(x) = x + √3 – x
y=________

#4. Consider the function below. (Round the answers to three decimal places. If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)
f(x) = 5 + 2×2 – x4
(a) Find the intervals of increase. (Enter the interval that contains smaller numbers first.)
( , ) ∪ ( , )

Find the intervals of decrease. (Enter the interval that contains smaller numbers first.)
( , ) ∪ ( , )

(b) Find the local minimum value.
Find the local maximum values.
(smaller x value)
(larger x value)

(c) Find the inflection points.
( , ) (smaller x value)
( , ) (larger x value)

Find the interval the function is concave up.
( , )

Find the intervals the function is concave down. (Enter the interval that contains smaller numbers first.)
( , ) ∪ ( , )

#5. Consider the following.
f(t) = t + cos(t)
-2π t 2π
(a) Find the interval(s) of increase. (Select all that apply.)

Find the interval(s) of decrease. (Select all that apply.)
(b) Find the local maximum value(s). (Select all that apply.)

Find the local minimum value(s). (Select all that apply.)
(c) Find the interval(s) where the graph is concave upward. (Select all that apply.)
Find the interval(s) where the graph is concave downward. (Select all that apply.)
Find the inflection point(s).
#6. Consider the function below. (If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

(a) Find the vertical and horizontal asymptotes.
x =
y =

(b) Find the interval where the function is increasing.
( , )

Find the intervals where the function is decreasing. (Enter the interval that contains smaller numbers first.)
( , ) ∪ ( , )

(c) Find the local minimum value.
(d) Find the inflection point.
( , )

Find the interval where the function is concave up. (Enter the interval that contains smaller numbers first.)
( , ) ∪ ( , )

Find the intervals where the function is concave down.
( , )

#8. (where t > 0) is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.01, p = 4, k = 0.05, and t is measured in minutes, estimate the times t corresponding to the inflection points. (Round your answers to two decimal places.)
t= __________ min (smaller value)
t = __________ min (larger value)

#21. Find a cubic function, in the form below, that has a local maximum value of 4 at -4 and a local minimum value of 0 at 2.
f (x) = ax3 + bx2 + cx + d
f (x) =_____________
#23. Consider the equation below. (Give your answers correct to two decimal places. If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

(a). Find the interval on which f is decreasing.
(____, _____)
(b) Find the local maximum value of f.
______

(c) Find the inflection point.
( , )

Find the interval on which f is concave up.
( , )

Find the interval on which f is concave down.
( , )

Section 5.3
#6. Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].

#7. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

y =
0 5 sin3 t dt

ex

Y’=
#9. Let g(x) =
x f(t)dt

−3
, where f is the function whose graph is shown.
(b) Estimate g(-2), g(-1), and g(0).
G(-2)=
G(-1)=
G(0)=

(c) On what interval is g increasing?
(____,_____)

(d) Where does g have a maximum value?
X=______

#11.
Evaluate the integral.

_____

#12. EXAMPLE 2 Find the derivative of the function below.

SOLUTION Since f(t) = √5 + t2 is continuous, Part 1 of the Fundamental Theorem of Calculus gives the answer.
G’(x)=__________
#14. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

g'(x) =__________
#15. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
g(y) =
y t2 sin 4t dt

6
G’(y)=_______

#17. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

G’(y)=_______

#18. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

G’(y)=_______
19. Find the derivative of the function.
#20. Let g(x) =
x f(t)dt

, where f is the function whose graph is shown.

(a) Evaluate g(x) for x = 0, 1, 2, 3, 4, 5, and 6.
g(0) =
g(1) =
g(2) =
g(3) =
g(4) =
g(5) =
g(6) =

(b) Estimate g(7). (Use midpoint to get the most precise estimate.)
g(7) =

(c) Where does g have a maximum and a minimum value?
minimum at x =
maximum at x =

#21. Consider the functions below.

#23. Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.
y = x−3, 3 ≤ x ≤ 7
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