Spring 2019 MATH105 Calculus I: Assignment 2 on Differentiation

This document presents the solutions to a Calculus I assignment (MATH105) from Spring 2019, focusing on differentiation techniques and their applications. The assignment includes problems on finding derivatives of various functions using rules like the quotient rule, product rule, and chain rule. It also covers finding equations of tangent lines to curves, solving related rates problems, implicit differentiation, logarithmic differentiation, and finding linearizations and differentials of functions. Specific problems involve finding derivatives of composite functions, trigonometric functions, logarithmic functions, and algebraic functions. Additionally, the assignment explores applications of differentiation, such as finding velocity and acceleration in a vibrating system and solving problems involving increasing areas and volumes. The solutions provide step-by-step guidance on how to approach and solve these calculus problems. Desklib offers more solved assignments and past papers for students.

Assignment No. 02 (Spring 2019) (Homework)
Mansour Mohammed Saeed Alkaabi
MATH105_20357_Spring_2019, section Calculus I (02), Spring 2019
Instructor: WebAssign Vendor
Current Score : 54 / 61 Due : Friday, March 1, 2019 11:59 PM +04
1. 1/1 points | Previous AnswersSCalcET8M 3.2.011.MI.
Differentiate.
34y2+21y4+5
2. 1/1 points | Previous AnswersSCalcET8M 3.2.031.
Find an equation of the tangent line to the given curve at the specified point.
WebAssign
Last Saved : n/a Saving... ()
F(y) = − (y + 5y 3)
1
y2
7
y4
y = , (1, 0)
x2 − 1
x2 + x + 1
y =
23x−23
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3. 2/2 points | Previous AnswersSCalcET8M 3.2.036.
(a) The curve is called a serpentine. Find an equation of the tangent line to this curve at the point
y =
−450x+2750
(b) Illustrate part (a) by graphing the curve and tangent line on the same screen.
4. 3/3 points | Previous AnswersSCalcET8M 3.2.043.
Suppose that Find the following values.
(a)
-18
(b)
-54/81
(c)
6
y = x/(1 + x 2) (3, 0.30).
f(5) = 3, f '(5) = 4, g(5) = −9, and g'(5) = 6.
(fg)'(5)
(f/g)'(5)
(g/f)'(5)
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5. 4/4 points | Previous AnswersSCalcET8M 3.2.052.
If f is a differentiable function, find an expression for the derivative of each of the following functions.
(a)
(b)
(c)
(d)
y = x7f(x)
y' = x6f '(x) − 6x 7f(x)
y' = x6f '(x) + 6x 7f(x)
y' = x7f(x) + 7x 6f '(x)
y' = x7f '(x) + 7x 6f(x)
y' = x7f(x) − 7x 6f '(x)
y = f(x)
x8
y' = xf(x) + 8f '(x)
x8
y' = xf '(x) + 7f(x)
x8
y' = xf '(x) − 7f(x)
x9
y' = xf(x) − 8f '(x)
x7
y' = xf '(x) − 8f(x)
x9
y = x3
f(x)
y' = f '(x)(3x 2) − x 3f(x)
[f(x)] 2
y' = f '(x)(2x 3) + x 2f(x)
[f(x)] 2
y' = f(x)(2x 3) + x 3f '(x)
[f(x)] 2
y' = f '(x)(2x 2) − x 3f '(x)
[f(x)] 2
y' = f(x)(3x 2) − x 3f '(x)
[f(x)] 2
y = 3 + xf(x)
x
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6. 1/1 points | Previous AnswersSCalcET8M 3.3.022.
Find an equation of the tangent line to the curve at the given point.
y =
5x+5
7. 3/3 points | Previous AnswersSCalcET8M 3.3.031.
(a) Use the Quotient Rule to differentiate the function
(b) Simplify the expression for by writing it in terms of sin(x) and cos(x), and then find
(c) Are your answers to parts (a) and (b) equivalent?
y' = xf '(x) − 2x 2f '(x) + 3
2x3/2
y' = xf(x) − 3x 2f '(x) + 3
2x3/2
y' = xf(x) + 3x 2f(x) − 2
2x3/2
y' = xf '(x) + 3x 2f(x) − 2
2x3/2
y' = xf(x) + 2x 2f '(x) − 3
2x3/2
y = 5ex cos(x), (0, 5)
f(x) = .
tan(x) − 1
sec(x)
f '(x) =
sec3(x)−tan2(x)sec(x)+tan(x)sec(x)sec2(x)
f(x) f '(x).
f '(x) =
cos(x)+sin(x)
Yes
No
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8. 5/6 points | Previous AnswersSCalcET8M 3.3.036.
An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then
released, it vibrates vertically. The equation of motion is where s is measured in centimeters and t in
seconds. (Take the positive direction to be downward.)
(a) Find the velocity and acceleration at time t.
(b) Graph the velocity and acceleration functions.
(c) When does the mass pass through the equilibrium position for the first time? (Round your answer to two decimal places.)
t = 2.27 s
(d) How far from its equilibrium position does the mass travel? (Round your answer to two decimal places.)
s = -0.03 cm
(e) When is the speed greatest? (Use n, a positive integer.)
s = 6 cos(t) + 5 sin(t), t ≥ 0,
v(t) =
−6sin(t) + 5cos(t)
a(t) =
−(6cos(t)+5sin(t))
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9. 2/2 points | Previous AnswersSCalcET8M 3.3.041.
Find the limit.
3
10.2/2 points | Previous AnswersSCalcET8M 3.4.002.
Write the composite function in the form [Identify the inner function and the outer function
Find the derivative dy/dx.
0.69 + n π
0.69 + 2n π
2.27 + n π
2.27 + 2n π
2.96 + 3n π
lim
t → 0
tan(9t)
sin(3t)
f(g(x)). u = g(x) y = f(u).]
y = 6 + 3x
(g(x), f(u)) =
6+3x,√u
=
32√6+3x
dy
dx
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11.1/1 points | Previous AnswersSCalcET8M 3.4.021.
Find the derivative of the function.
12.1/1 points | Previous AnswersSCalcET8M 3.4.042.
Find the derivative of the function.
y = sin(sin(sin(x)))
y' =
cos(sin(sin(x)))cos(sin(x))cos(x)
y = x
x + 5
y' =
52√x(x+5)(32)
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13.2/2 points | Previous AnswersSCalcET8M 3.4.055.
(a) Find an equation of the tangent line to the curve at the point (0, 3).
y=32x+3
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
14.1/1 points | Previous AnswersSCalcET8M 3.4.076.
For what values of r does the function satisfy the differential equation (Enter your answers as a comma-
separated list.)
y = 6/(1 + e −x)
y = erx y'' − 6y' + 2y = 0?
r =
3−√7,3+√7
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15.2/2 points | Previous AnswersSCalcET8M 3.5.009.
Find by implicit differentiation.
16.1/1 points | Previous AnswersSCalcET8M 3.5.018.
Find dy/dx by implicit differentiation.
17.2/2 points | Previous AnswersSCalcET8M 3.5.503.XP.MI.
Find dy/dx by implicit differentiation.
18.1/1 points | Previous AnswersSCalcET8M 3.6.015.
Differentiate the function.
dy/dx
= y2 + 1x2
x + y
y' =
x(x+2y)2y(x+y)2+x2
x sin(y) + y sin(x) = 9
y' =
−(ycos(x )+sin(y))(xcos(y)+ sin(x))
x8 + y7 = 5
y' =
−8x77y6
F(s) = ln(ln(7s))
F '(s) =
1sln(7s)
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19.1/1 points | Previous AnswersSCalcET8M 3.6.020.
Differentiate the function.
20.2/2 points | Previous AnswersSCalcET8M 3.6.027.
Differentiate f and find the domain of f. (Enter the domain in interval notation.)
derivative =
2x−ln(x−8)(x−8)−8(x−8)(1−ln(x−8 ))2
domain
(8,e+8)∪(8+e,∞)
H(z) = ln b2 − z2
b2 + z2
H'(z) =
−2b2z(z2+b2)(−z2+b2)
f(x) = x
1 − ln(x − 8)
f '(x)
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21.3/3 points | Previous AnswersSCalcET8M 3.6.036.
Find the equation of the tangent line to the curve at the points
Illustrate by graphing the curve and its tangent lines.
y = (9 ln(x))/x (1, 0) and (e, 9/e).
at the point (1, 0) y =
9x−9
at the point (e, 9/e) y =
9e
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22.1/1 points | Previous AnswersSCalcET8M 3.6.040.
Use logarithmic differentiation to find the derivative of the function.
y' =
ex2(x2+2)112√x+√x(2ex2x(x2+2)11+22ex2x(x2+2)10)
23.1/1 points | Previous AnswersSCalcET8M 3.6.044.MI.
Use logarithmic differentiation to find the derivative of the function.
y' =
x2cos(x)(2cos(x)x−2ln(x)sin(x))
24.2/2 points | Previous AnswersSCalcET8M 3.9.002.
(a) If A is the area of a circle with radius r and the circle expands as time passes, find in terms of
(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate
of 1 m/s, how fast is the area of the spill increasing when the radius is
26·2·π
m2/s
25.1/1 points | Previous AnswersSCalcET8M 3.9.003.
Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is
16 cm2?
48 cm2/s
y = ex2(x2 + 2) 11x
y = x2 cos(x)
dA/dt dr/dt.
=
2πr
dA
dt
dr
dt
26 m?
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26.1/1 points | Previous AnswersSCalcET8M 3.9.005.MI.
A cylindrical tank with radius 7 m is being filled with water at a rate of 2 m 3/min. How fast is the height of the water increasing?
249π
m/min
27.2/2 points | Previous AnswersSCalcET8M 3.9.009.
Suppose where x and y are functions of t.
(a) If find when
(b) If find when
28.–/2 pointsSCalcET8M 3.9.033.
The top of a ladder slides down a vertical wall at a rate of 0.125 m/s. At the moment when the bottom of the ladder is 5 m from the
wall, it slides away from the wall at a rate of 0.3 m/s. How long is the ladder?
m
29.1/1 points | Previous AnswersSCalcET8M 3.10.003.
Find the linearization of the function at a.
y = ,2x + 1
dx/dt = 6, dy/dt x = 4.
= 2
dy
dt
dy/dt = 2, dx/dt x = 24.
= 14dx
dt
L(x)
f(x) = , a = 9x
L(x) =
x+96
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30.0/2 points | Previous AnswersSCalcET8M 3.10.013.
Find the differential of each function.
(a)
(b)
31.3/3 points | Previous AnswersSCalcET8M 3.10.019.
Compute Δy and dy for the given values of x and
Sketch a diagram showing the line segments with lengths
y = tan( )7t
dy =
$$√7tsec2(√7t)2t
y = 2 − v2
2 + v2
dy =
−8v(2+v2)2
dx = Δx.
y = x2 − 4x, x = 3, Δx = 0.5
Δy = 1.25
dy = 1
dx, dy, and Δy.
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32.1/1 points | Previous AnswersSCalcET8M 3.10.501.XP.MI.
Find the linearization L(x) of the function at a.
33.0/2 points | Previous AnswersSCalcET8M 3.10.503.XP.
Find the differential of each function.
(a)
(b)
Enhanced Feedback
Please try again. Keep in mind that the differential of is defined as Make sure you apply the Product Rule
when f is a product and use the Chain Rule when f is a composition of functions.
f(x) = x 4 + 4x2, a = 1
L(x) =
12x−7
y = x2 sin(8x)
dy =
2xsin(8x)+8x2cos(8x)
y = ln 3 + t2
dy =
t3+t2
y = f(x) dy = f '(x) dx.
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