Problem Sets: Applications of Derivatives (导数的应用)

Last Update: 2025-10-14

Remark:

• Per the sequence of the course outline, this set does not yet cover problems related to exponential functions, inverse functions, and logarithms—these will be included in subsequent sets.
• 根据课程大纲的内容顺序安排，此部分练习暂未涵盖与指数函数、反函数及对数函数相关的题目 —— 这类题目将在后续练习中纳入。

• If the equation \(2y^{3} - 2y^{2} + 2xy - x^{2} = 1\) defines \(y= y(x)\) implicitly, please find the extreme value of \(y= y(x)\).

• If the equation \(x^{2/3} + y^{2/3} = 1\) defines \(y= y(x)\) implicitly, please find the extreme value of \(y= y(x)\).

• If \( f(x) = k(x^{2} - 3)^{2} \) where \( k \) is a constant. Find the value of \( k \) such that the normal line to the curve at the point of inflection is through the origin \( (0,0) \).

• Suppose \( f(x) \) is continuous, and \(\lim_{x \to 0} \dfrac{f(x) - \lvert x\rvert}{x^{2}} = -1\).
  • Is \( f(x) \) is differentiable at \( x = 0 \)?
  • Is \( f(0) \) is an extreme value?
• Sketch the graph of the function \(y = f(x)\)
  • \(f (x) = \dfrac{x^{2}}{\sqrt{x+1}}\)
  • \(f(x) = x^{2/3}(6-x)^{1/3}\)
  • \(f(x) = \dfrac{x^{3} +4}{x^{2}}\)
  • \(f(x) = \dfrac{x\lvert x\rvert}{1+x}\)

• Find the highest and lowest points on the curve \( x^{2} + xy + y^{2} = 12 \).

• A differentiable function \( y = y(x) \) is implicitly defined by the equation \(3x^{2} + 2xy + 3y^{2} = 1\)
  • Find all the extreme values of \( y = y(x) \).
  • In the first quadrant (\( x \geq 0, y \geq 0 \)), the tangent line of \( y = y(x) \) intersects the \( x \)-axis and \( y \)-axis at the point \( A \) and \( B \) respectively. Find the minimal area of the triangle \( \Delta ABO \) connecting the points \( A \), \( B \) and the origin \( O \).

• If the equation \( y^{3} + xy^{2} + x^{2}y + 6 = 0 \) defines \( y = y(x) \) implicitly, please find the extreme value of \( y(x) \).

• Show that \( (x^{2} - x)(4 - y^{2}) \leq 16 \) for all numbers \( x \) and \( y \) such that \( \lvert x\rvert \leq 2 \) and \( \lvert y\rvert \leq 2 \).

• Find the point on the parabola \( y = 1 - x^{2} \) at which the tangent line cuts from the first quadrant the triangle with the smallest area.

• Show that the inflection points of the curve \( y = (\sin x)/x \) lie on the curve \( xy^{2} + 4 = 4 \).

• Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely?

• An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point \(0,a)\) on the \(y\)-axis. Find the value of a that minimizes the lengths of the equal sides.

• Find the area of the largest rectangle that can be inscribed in a semicircle of radius \(r\).

• The line \(y = mx + b\) intersects the parabola \(y = x^{2}\) in points \(A\) and \(B\). Find the point \(P\) on the arc \(AOB\) of the parabola that maximizes the area of the triangle \(PAB\).