Calc 2 Midterm 2 Problems
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 400g, 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 + 11.5t, 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
