U - Sub
Identify the correct substitution u and du for the integral:
∫ 2x√(x² + 1) dx
Let u = x² + 1, then du = 2x dx.
What is the first step when solving a related rates problem?
Draw a diagram and identify all variables.
What is ∫ e^x · cos(x) dx?
Use integration by parts twice.
Result: (e^x)(sin x + cos x)/2 + C
What is the derivative of x²?
2x
What is the sum of a geometric series with a = 2 and r = 1/2?
Sum = a / (1 - r) = 2 / (1 - 1/2) = 4
Evaluate the integral using substitution:
∫ x·cos(x²) dx
∫ x·cos(x²) dx = (1/2)∫cos(u) du = (1/2)·sin(x²) + C.
A balloon rises at 5 ft/sec. How fast is the distance from a person 60 ft away increasing when the balloon is 80 ft high?
Use Pythagorean Theorem:
Let z² = x² + y². Then dz/dt = (y/z)·dy/dt = (80/100)·5 = 4 ft/sec.
Solve ∫ ln(x) dx.
Use integration by parts:
Let u = ln(x), dv = dx →
∫ ln(x) dx = x·ln(x) - x + C.
What is the derivative of sin(x)·cos(x)?
Use product rule:
cos²(x) - sin²(x) = cos(2x)
Does the series ∑(1/n²) converge or diverge?
Converges (p-series with p = 2 > 1)
Use substitution to evaluate the integral:
∫ (3x²)/(x³ + 1) dx
Let u = x³ + 1, then du = 3x² dx.
∫ (3x²)/(x³ + 1) dx = ∫ du/u = ln|x³ + 1| + C.
Water fills a cone at 2 cm³/sec. Radius = half the height. Find dh/dt when h = 4 cm.
Volume = (1/3)πr²h with r = h/2.
So V = (π/12)h³.
Then dV/dt = (π/4)h²·dh/dt. Solve for dh/dt.
Evaluate ∫ sec³(x) dx.
Answer: (1/2)·sec(x)·tan(x) + (1/2)·ln|sec(x) + tan(x)| + C.
Find d/dx of ln(x² + 1).
(2x)/(x² + 1)
What is the radius of convergence of the series ∑(xⁿ / n!)?
Radius of convergence R = ∞ (entire function)
Use substitution to evaluate the following integral:
∫ (x² + 1)^3 · 2x dx
Let u = x² + 1, then du = 2x dx.
∫ (x² + 1)^3 · 2x dx = ∫ u³ du = (1/4)(x² + 1)^4 + C.
A 10-ft ladder slides down a wall. If the base is moving away at 1 ft/sec, how fast is the top sliding down when the base is 6 ft from the wall?
Use x² + y² = 100. Differentiate:
2x(dx/dt) + 2y(dy/dt) = 0.
Then dy/dt = -(x/y)·dx/dt = -(6/8)·1 = -0.75 ft/sec.
What method would you use to solve ∫ 1 / (x² + 4) dx?
Use trig substitution or recognize arctangent form.
Result: (1/2)·arctan(x/2) + C.
What is the second derivative of f(x) = e^(2x)?
f'(x) = 2e^(2x), f''(x) = 4e^(2x)
What test determines if the series ∑(-1)ⁿ / n converges?
Alternating Series Test (Leibniz Test)
Evaluate the integral using substitution:
∫ x / √(1 + x²) dx
Let u = 1 + x², then du = 2x dx.
∫ x / √(1 + x²) dx = (1/2)∫ du/√u = √(1 + x²) + C.
Two cars leave the same point, one east at 30 mph, one north at 40 mph. How fast is the distance between them changing after 2 hours?
After 2 hours: x = 60, y = 80, z = 100.
dz/dt = (1/z)(x·dx/dt + y·dy/dt) = (1/100)(60·30 + 80·40) = 50 mph.
Evaluate ∫ x²·e^x dx.
Use integration by parts twice.
Answer: x²·e^x - 2x·e^x + 2e^x + C.
Use implicit differentiation to find dy/dx if x² + y² = 25.
Differentiate both sides:
2x + 2y(dy/dx) = 0 → dy/dx = -x/y
Find the interval of convergence for the series ∑(xⁿ / n).
Use Ratio Test:
|x| < 1 → Check endpoints:
At x = 1 → Harmonic series diverges
At x = -1 → Alternating harmonic converges
Interval: (-1, 1]