Calculus Notes and Exercises
Introduction to Derivatives
Notes:
Definition: The derivative of a function f at a point x is given by:
f'(x) = lim_{h \to 0} (f(x+h) - f(x))/h
Interpretation: The derivative represents the rate of change or the slope of the tangent line
to the function at a point.
Basic Rules:
Power Rule: d/dx x^n = nx^{n-1}
Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Quotient Rule: d/dx [f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x))/g(x)^2
Chain Rule: d/dx f(g(x)) = f'(g(x))g'(x)
Exercises:
1. Exercise 1: Find the derivative of f(x) = 3x^4 + 2x^2 - x + 7.
Solution: f'(x) = 12x^3 + 4x - 1
2. Exercise 2: Differentiate g(x) = (5x^3 - 2x + 1)/x^2.
Solution: g'(x) = (15x^2 - 2x^2 - 10x + 4)/(x^4) = 15 - 10/x - 2/x^2 + 4/x^3
3. Exercise 3: If h(x) = sin(x^2), find h'(x).
Solution: h'(x) = 2x cos(x^2)
Applications of Derivatives
Notes:
Critical Points: Points where f'(x) = 0 or f'(x) does not exist.
Increasing/Decreasing Functions: If f'(x) > 0 on an interval, f is increasing. If f'(x) < 0, f is
decreasing.
Concavity and Inflection Points:
, f is concave up if f''(x) > 0.
f is concave down if f''(x) < 0.
An inflection point is where f changes concavity.
Exercises:
4. Exercise 1: Find the critical points of f(x) = x^3 - 3x^2 + 4x - 2.
Solution: f'(x) = 3x^2 - 6x + 4
Set f'(x) = 0:
3x^2 - 6x + 4 = 0
x=1
Critical point: x = 1
5. Exercise 2: Determine the intervals where g(x) = x^4 - 4x^2 is increasing or decreasing.
Solution: g'(x) = 4x^3 - 8x = 4x(x^2 - 2)
Set g'(x) = 0:
4x(x^2 - 2) = 0
x = 0, ±sqrt(2)
Intervals:
(-∞, -sqrt(2)): g'(x) < 0
(-sqrt(2), 0): g'(x) > 0
(0, sqrt(2)): g'(x) < 0
(sqrt(2), ∞): g'(x) > 0
6. Exercise 3: Identify the inflection points of h(x) = x^4 - 4x^3 + 6x^2.
Solution: h''(x) = 12x^2 - 24x + 12
Set h''(x) = 0:
12x^2 - 24x + 12 = 0
x=1
Inflection point: x = 1
Introduction to Integrals
Notes:
Definition: The integral of a function f over an interval [a, b] is given by:
∫_a^b f(x) dx
Interpretation: The integral represents the area under the curve of f from a to b.
Basic Rules:
Power Rule: ∫ x^n dx = x^{n+1}/(n+1) + C (for n ≠ -1)
Introduction to Derivatives
Notes:
Definition: The derivative of a function f at a point x is given by:
f'(x) = lim_{h \to 0} (f(x+h) - f(x))/h
Interpretation: The derivative represents the rate of change or the slope of the tangent line
to the function at a point.
Basic Rules:
Power Rule: d/dx x^n = nx^{n-1}
Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Quotient Rule: d/dx [f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x))/g(x)^2
Chain Rule: d/dx f(g(x)) = f'(g(x))g'(x)
Exercises:
1. Exercise 1: Find the derivative of f(x) = 3x^4 + 2x^2 - x + 7.
Solution: f'(x) = 12x^3 + 4x - 1
2. Exercise 2: Differentiate g(x) = (5x^3 - 2x + 1)/x^2.
Solution: g'(x) = (15x^2 - 2x^2 - 10x + 4)/(x^4) = 15 - 10/x - 2/x^2 + 4/x^3
3. Exercise 3: If h(x) = sin(x^2), find h'(x).
Solution: h'(x) = 2x cos(x^2)
Applications of Derivatives
Notes:
Critical Points: Points where f'(x) = 0 or f'(x) does not exist.
Increasing/Decreasing Functions: If f'(x) > 0 on an interval, f is increasing. If f'(x) < 0, f is
decreasing.
Concavity and Inflection Points:
, f is concave up if f''(x) > 0.
f is concave down if f''(x) < 0.
An inflection point is where f changes concavity.
Exercises:
4. Exercise 1: Find the critical points of f(x) = x^3 - 3x^2 + 4x - 2.
Solution: f'(x) = 3x^2 - 6x + 4
Set f'(x) = 0:
3x^2 - 6x + 4 = 0
x=1
Critical point: x = 1
5. Exercise 2: Determine the intervals where g(x) = x^4 - 4x^2 is increasing or decreasing.
Solution: g'(x) = 4x^3 - 8x = 4x(x^2 - 2)
Set g'(x) = 0:
4x(x^2 - 2) = 0
x = 0, ±sqrt(2)
Intervals:
(-∞, -sqrt(2)): g'(x) < 0
(-sqrt(2), 0): g'(x) > 0
(0, sqrt(2)): g'(x) < 0
(sqrt(2), ∞): g'(x) > 0
6. Exercise 3: Identify the inflection points of h(x) = x^4 - 4x^3 + 6x^2.
Solution: h''(x) = 12x^2 - 24x + 12
Set h''(x) = 0:
12x^2 - 24x + 12 = 0
x=1
Inflection point: x = 1
Introduction to Integrals
Notes:
Definition: The integral of a function f over an interval [a, b] is given by:
∫_a^b f(x) dx
Interpretation: The integral represents the area under the curve of f from a to b.
Basic Rules:
Power Rule: ∫ x^n dx = x^{n+1}/(n+1) + C (for n ≠ -1)
