Calc 2 Midterm 2 Problems

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Verify the following identity using the definitions of the hyperbolic functions. cosh(x) + sinh(x) = e^(x)

((e^(x) + e^(-x))/2) + ((e^(x) - e^(-x))/2)

Evaluate the derivative of the following function. f(x) = (6x)^(6x)

(6x)^(6x)*6(ln(6x) + 1)

7.3 ||| Hyperbolic Function Cosh(t) Definition

(e^(t) + e^(-t))/2

7.3 ||| Hyperbolic Function Sinh(t) Definition

(e^(t) - e^(-t))/2

Use substitution to find the indefinite integral ∫ 64/(8x + 5)^(9) dx

- 1/(8x+5)^(8) + C

A​ heavy-duty shock absorber is compressed 4cm from its equilibrium position by a mass of 400 kg. How much work is required to compress the shock absorber 6cm from its equilibrium​ position? (A mass of 400kg exerts a force​ (in newtons) of 400​g, where g≈9.8m/s2​)

-0.06 ∫ (98000y) dy 0

FInd the indefinite integral ∫ (4x-1)^(-9) dx

-1/32 (4x-1)^(-8) + C

Evaluate the following integral. ∫ e^(5x)/(e^(5x) + 7) dx

-1/5 ln|e^(5x) + 7| + C

Find the indefinite integral. ∫ e^(1-8t) dt

-1/8 e ^(1-8t) + C

Compute the following derivative using the method of your choice. d/dx (e^(-9x^(2)))

-18xe^(-9x^(2))

Find the derivative of the following function. f(x) = -sinh^(6)(4x)

-6 * sinh^(5)(4x) * cosh(4x) * 4

Devise the exponential growth function that fits the given​ data, then answer the accompanying question. Be sure to identify the reference point ​(t=​0) and units of time. ​Uranium-238 (U-238) has a​ half-life of 4.5 billion years. Geologists find a rock containing a mixture of​ U-238 and​ lead, and determine that 79​% of the original​ U-238 remains; the other 21​% has decayed into lead. How old is the​ rock?

.79 y(0) = y₀e^(-0.154t)

Let R be the region bounded by: y = x^(2)​, x = ​1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the line x = −9.

1 ∫ 2π(x+9)(x^(2)) dx 0

Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the​ y-axis. y = 12x, y = 12, and x = 0

1 ∫ 2πx(12-12x) dx 0

find the derivative of the following function. f(x) = ln(csch(11x))

1/(csch(11x)) * -csch(11x) * coth(11x) * 11

Evaluate the following integral. ∫ e^(6x)/(e^(6x) - 8e^(-6x)) dx

1/12 ln|e^(12x) - 8| + C

Evaluate ∫ sinh(15x) dx

1/15 cosh(15x) = C

Evaluate the integral. e^(4√(t)) / √(t) dt

1/2 e^(4√(t)) + C

Evaluate the following integral. e^(5x) / (e^(5x) + 3) dx

1/5 ln|e^(5x) + 3| + C

Evaluate the following integral. e^(10) ∫ (ln(x))^(2) / x dx 1

10 ∫ u^(2) du 0

Find the derivative of the following function. f(x) = sinh(10x)

10cosh(10x)

Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the​ x-axis. y = √(x), y = 0, x = 121

11 ∫ 2πy(121 - y^(2)) dy 0

Find the area of the surface generated when the given curve is revolved about the given axis. y = 8x - 6 for [11/8, 21/8] about the Y-axis

15 ∫ 2π((y+6)/8)√(1 + 1/64) dy 5

Find the area of the surface generated when the given curve is revolved about the given axis. y = 4x + 3 for [0, 2], about the X-axis

2 ∫ 2π(4x + 3)√(1 + 16) dx 0

Find the area of the surface generated when the given curve is revolved about the given axis. y = (5x)^(1/3) for [0, 8/5] about the Y-axis

2 ∫ 2π(y^(3)/5)√(1+ (9/25)y^(4)) dy 0

Find the mass of the thin bar with the given density function. p(x) = { 5 if [0,2] { 5 + x if (2, 4]

2 4 ∫ 5 dx + ∫ (5 + x) dx 0 2

Evaluate the integral 3 ∫ 2^(x) dx -3

2^(x)/ln(3) + C | 3 | -3

Find the mass of the thin bar with the given density function. p(x) = x√(10 - x^(2)) for [0, 3]

3 ∫ x√(10-x^(2))dx 0

Evaluate the following integral. 4e ∫ (3^(ln(x))) / x dx 1

3^(u) / ln(3) + C | ln(4) + 1 | 0

How much work is required to move an object from x=1 to x=4 ​(measured in​ meters) in the presence of a force​ (in N) given by F(x)=2/x^(2) acting along the​ x-axis?

4 ∫ 2/(x^2) dx 1

Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the​ y-axis. y = 4x - x^(2), y = 0

4 ∫ 2πx(4x - x^(2)) dx 0

Find the arc length of the curve below on the given interval. y = (1/4)x^(4) + 1/(8x^(2)) on [1, 4]

4 ∫ √(x^(6) + 1/2 + (1/16)x^(-6)) dx 1

Find the arc length of the curve below on the given interval. x = (1/4)y^(4) + 1/(8y^2) on [2, 4]

4 ∫ √(y^(6) + 1/2 + 1/(16x^(6))) dy 2

Evaluate the derivative of the following function. f(x) = (4x)^(4x)

4 * e^(4xln(4x)) * (ln(4x) + 1)

Evaluate the following integral. ∫ x^(2) * 4^(x^(3) + 5) dx

4^(x^(3) + 5) / (3ln(4)) + C

Evaluate the following integral. 4 ∫ (1 + ln(x))x^(x) dx 1

4ln(4) ∫ e^(u) du 0 u = xln(x) - Don't plug this in!

Find the derivative of the following function. f(x) = x^(4)sinh^(5)(5x)

4x^(3)*sinh^(5)(5x) + x^(4)*5sinh^(4)(5x)*cosh(5x)*5

Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the​ x-axis. y = 5 - x, y = 3, and x = 0

5 ∫ 2πy(5-y) dy 3

e^(5) ∫ (ln^(2)(x^(2)))/x dx 1

5 ∫ u^(2) du 0

Find the arc length of the curve below on the given interval. y = 2x^(3/2) on [0, 7]

7 ∫ √(1+9x) dx 0

Use the shell method to find the volume of the solid formed when a hole of radius 4is drilled symmetrically along the axis of a right circular cone of radius 8 and height 12.

8 ∫ 2πx(-(3/2)x + 12) dx 4

Find the arc length of the curve below on the given interval. y = (3/4)x^(4/3) - (3/8)x^(2/3) + 6 on [1, 8]

8 ∫ √(x^(2/3) + 1/2 + (1/16)x^(-2/3)) dx 1

Find the area of the surface generated when the given curve is revolved about the given axis. y = 8√(x) for [65, 84] about the X-axis

84 ∫ 2π(8√(x))√(1+16x^(-1)) dx 65

Find the arc length of the curve below on the given interval. y = 1/27(9x^(2) + 6)^(3/2) on [3, 9]

9 ∫ √(9x^(4) + 6x^(2) + 1) dx 3

In the year​ 2000, the population of a certain country was 282 million with an estimated growth rate of 0.6​% per year. a. Based on these​ figures, find the doubling time and project the population in 2100. b. Suppose the actual growth rates are just 0.2 percentage points lower and higher than 0.6​% per year ​(0.4​% and 0.8​%). What are the resulting doubling times and projected 2100 ​population?

A) Doubling Time --- 2 = e^(0.006t) 2100 Population --- y(100) = 282e^(0.006 * 100) B) 0.4% Growth Rate Doubling Time --- 2 = e^(0.004t) 2100 Population --- y(100) = 282e^(0.004*100) 0.8% Growth Rate Doubling Time --- 2 = e^(0.008t) 2100 Population --- y(100) = 282e^(0.008*100)

Calculate the work required to stretch the following springs 0.5 m from their equilibrium positions. Assume​ Hooke's law is obeyed. a. A spring that required a force of 40 N to be stretched 0.1 m from its equilibrium position. b. A spring that required 50 J of work to be stretched 0.2 m from its equilibrium position.

A) 0.5 ∫ (400x) dx 0 B) 0.5 ∫ (2500x) dx 0

Evaluate the derivative of the following function. h(x) = 3^(x^(3))

ln(3)(3^(x)^(3))(3x^(2))

Determine the indefinite integral. ∫ sinh(x)/(cosh(x) + 2) dx

ln|cosh(x) + 2| + C

Find the following integral. (e^(x) + e^(-x)) / (e^(x) - e^(-x)) dx

ln|e^(x) - e^(-x)| + C

Verify the identity using the definitions of the hyperbolic functions. tanh(x) = (e^(2x) - 1)/(e^(2x) + 1)

sinh(x)/cosh(x)

7.2 ||| Exponential Growth Formula

y(t) = y₀e^(kt)

Find the mass of the thin bar with the given density function. p(x) = 1 + sin(x) for [0, π/6]

π/6 ∫ (1 + sin(x)) dx 0

Evaluate the following integral. ∫ 6^(-2x) dx

∫ -1/(36^(x)ln(36)) + C

7.1 ||| Derivative of an Exponential Function Formula

d/dx (b^(u(x))) = ln(b) * e^(u(x)) * (u'(x))

Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume​ Hooke's law is obeyed. a. A spring that required 120 J of work to be stretched 0.4 m from its equilibrium position. b. A spring that required a force of 200 N to be stretched 0.5 m from its equilibrium position.

A) 1.25 ∫ (1500x) dx 0 B) 1.25 ∫ (400x) dx 0

Suppose a force of 30N is required to stretch and hold a spring 0.1m from its equilibrium position. a. Assuming the spring obeys​ Hooke's law, find the spring constant k. b. How much work is required to compress the spring 0.2 m from its equilibrium​ position? c. How much work is required to stretch the spring 0.6 m from its equilibrium​ position? d. How much additional work is required to stretch the spring 0.1 m if it has already been stretched 0.1 m from its​ equilibrium?

A) 30 = K * 0.1 B) -0.2 ∫ (300x) dx 0 C) 0.6 ∫ (300x) dx 0 D) 0.2 ∫ (300x) dx 0.1

Devise an exponential decay function that fits the given​ data, then answer the accompanying questions. Be sure to identify the reference point ​(t=​0) and units of t. The pressure of a certain​ planet's atmosphere at sea level is approximately 800 millibars and decreases exponentially with elevation. At an elevation of 35,000 ​ft, the pressure is​ one-third of the​ sea-level pressure. At what elevation is the pressure half of the​ sea-level pressure? At what elevation is it 3​% of the​ sea-level pressure?

A) 400 = 800e^(-3.138892*10^(-5) * t) B) 24 = 800e^(-3.138892*10^(-5) * t)

A spring on a horizontal surface can be stretched and held 0.3 m from its equilibrium position with a force of 33 N. a. How much work is done in stretching the spring 5.5 m from its equilibrium​ position? b. How much work is done in compressing the spring 1.5 m from its equilibrium​ position?

A) 5.5 ∫ (110x) dx 0 B) -1.5 ∫ (110x) dx 0

Two functions f and g are given. Show that the growth rate of the linear function is constant and that the relative growth rate of the exponential function is constant. ​f(t) = 55 + 11.5​t, g(t) = 55e^(t/11)

A) Growth Rate = y'(x) 11.5 B) Growth Rate = y'(x)/y(x) (5e^(t/11))/(55e^(t/11))

A quantity increases according to the exponential function y(t)=y₀e^(kt). What is the time required for the quantity to multiply by five​? What is the time required for the quantity to increase​ p-fold?

A) ln(5)/k B) ln(p)/k

A certain drug is eliminated from the bloodstream with a​ half-life of 36 hours. Suppose that a patient receives an initial dose of 60 mg of the drug at midnight. a. How much of the drug is in the​ patient's blood at noon later that​ day? b. When will the drug concentration reach 35​% of its initial​ level?

A) y(12) = 60e^(-0.019254 * 12) B) 21 = 60e^(-0.019254t)

8.1 ||| Review of Integration

Chain Rule + Product Rule + Exponential Functions

6.7 ||| Hooke's Law

F(x) = K*x

6.4 ||| Shell Method Formula

b ∫ 2π(x + d) (f(x) - g(x)) dx a

6.6 ||| Area of 3D Surface Formula

b ∫ 2πR(x)√(1 + (f'(x))^(2)) dx a

6.7 ||| Work Formula

b ∫ F(x) dx a

6.7 ||| Mass Formula

b ∫ p(x) dx a

6.5 ||| Arc Length Formula

b ∫ √(1 + (f'(x))^(2)) dx a

7.1 ||| Integral of an Exponential Function Formula

d/dx (b^(u(x))) = b^(u(x)) / ln(b) + C


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