10 Questions
What is the nature of the function f5 on the interval [−5, −3]?
Strictly increasing
What is the x-value of the strict local minimum of the function f6?
2
What is the nature of the function f7 on the interval (−∞, 8)?
Strictly decreasing
What is the nature of the function f8 on the interval (−∞, 1)?
Strictly increasing
What is the nature of the function f1?
Weakly concave up and weakly concave down
What is the nature of the function f2?
Strictly concave up
What is the nature of the function f3 on the interval (−∞, 7)?
Strictly concave down
What is the x-value of the inflection point of the function f3?
7
What is the nature of the function f4?
The second derivative is a linear function
What is the nature of the function f8 on the interval (1, +∞)?
Strictly decreasing
Study Notes
Derivatives and Tangent Lines
- f1(x) = 35x - 6, f1'(x) = 35, f1'(1) = 23, L1(x) = 23(x - 1) - 7
- f2(x) = -6x^4, f2'(-2) = -8, L2(x) = 8(x + 2) - 4
- f3(x) = √x, f3(1/4) = 2, L3(x) = 2x - 4 + 1
- f4(x) = -12x + 34, f4'(2) = 10, L4(x) = 10(x - 2) + 4
- f5(x) = 2x^2 + 16x + 32, f5'(-3) = -13, L5(x) = -13(x + 3) + 4
- f6(x) = -6 ln(7) * 7^x, f6(0) = -6 ln(7), L6(x) = -6 ln(7) x + 41
- f7(x) = -12(2 - 3x)^3, f7'(2) = 768, L7(x) = 768(x - 2) + 256
- f8(x) = (2x - 2)e^(x - 2)^(-15), f8'(-3) = -8, L8(x) = -8(x + 3) + 1
- f9(x) = 2e^(2x) ln(3x - 5) + e^(2x) / (3x - 5), f9'(2) ≈ 163.79, L9(x) = 163.79(x - 2)
- f10(x) = (10x + 6x)(8x^3 - 9x^2 + 100) + (5x^2 + 6x - 7)(24x^2 - 18x), f10'(-2) = 132, L10(x) = 132(x + 2)
Monotonicity and Extreme Values
- f1(x) = -8x, f1'(x) = -8 < 0 everywhere, f1 strictly decreases everywhere
- f2(x) = x^2 - 6x + 42, f2'(x) = 2x - 6, f2 strictly decreases on (-∞, 3], strictly increases on [3, +∞)
- f3(x) = x^3 - 243x, f3'(x) = 3x^2 - 243, strictly increases on (-∞, -9], strictly decreases on [-9, 9], strictly increases on [9, +∞)
- f4(x) = x^3 + 3x^2 + 3x + 100, f4'(x) = 3x^2 + 6x + 3 ≥ 0, f4 strictly increases on ℝ
- f5(x) = -3x^4 - 32x^3 - 90x^2, f5'(x) = -12x^3 - 96x^2 - 180x, f5 strictly decreases on ℝ
Convexity and Inflection Points
- f1(x) = 21/x + 7, f1''(x) = 0, f1 is everywhere concave up and concave down
- f2(x) = x^2 - 6x + 42, f2''(x) = 2 > 0, f2 is strictly concave up everywhere
- f3(x) = x^3 - 21x^2, f3''(x) = 6x - 42, f3 is strictly concave down on (-∞, 7], strictly concave up on [7, +∞), x = 7 is an inflection point
- f4(x) = x^3 + 3x^2 + 3x + 100, f4''(x) = 6x + 6
Solve various calculus problems involving derivatives, functions, and graphs. Practice your skills and test your knowledge with these exercises.
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