QCEVault

Further applications of differentiation — Question 332

Original QCE Vault practice · 8 marks

Q332 · Practice questionTechnology-activeComplex unfamiliar8 marks

QUESTION 332 (8 marks)

A rectangular logo has its lower edge on the xx-axis, left edge on the yy-axis and upper-right corner on y=e−xy=e^{-x}. Its width is b>0b>0 units.
Decreasing curve y=exp(-x). A rectangle from x=0 to x=b has its upper-right corner on the curve.
a)
Determine the width and area of the largest possible logo.
[3 marks]
b)
For that width, determine the exact area below the curve but outside the rectangle, between x=0x=0 and x=bx=b.
[2 marks]
c)
Determine the positive width for which the rectangle occupies exactly half the area under the curve over its width. Give the width to three decimal places.
[3 marks]
Question linkSyllabus coverage

Related questions

  1. Q3 · Original practice · 1 mark
    The graph shown is the graph of f′(x)f'(x). At which value of xx does ff have a local maximum?
    Further applications of differentiation
  2. Q13 · Original practice · 1 mark
    The stationary points of f(x)=x3−3x2−9x+4f(x)=x^3-3x^2-9x+4 occur when x=x=
    Further applications of differentiation
  3. Q23 · Original practice · 1 mark
    For f(x)=xe−xf(x)=xe^{-x} on x>0x>0, the xx-coordinate of the maximum point is
    Further applications of differentiation
  4. Q28 · Original practice · 4 marks
    A rectangle is inscribed symmetrically under the parabola y=12−x2y=12-x^2 and above the xx-axis, as shown. The upper right corner is (x,12−x2)(x,12-x^2), where 0<x<120<x<\sqrt{12}. Determine the maximum possible area of the rectangle.
    Further applications of differentiation