Question 1
For each function, sketch the gradient function \(f'(x)\) on a separate set of axes:
(a)\(f(x)=x^3-3x\)
(b)\(f(x)=e^{-x}\)
(c)\(f(x)=\sin x\) over \([0,2\pi]\)
(a)
Differentiate
\(f'(x)=3x^2-3\): upward parabola with roots at \(x=\pm1\), vertex at \((0,-3)\)
(b)
Differentiate
\(f'(x)=-e^{-x}\): always negative, approaching 0 as \(x o\infty\)
(c)
Differentiate
\(f'(x)=\cos x\): same shape as \(\cos x\)
Question 2
A graph of \(f'(x)\) is shown — a parabola with vertex \((1,-4)\) crossing the \(x\)-axis at \(x=-1\) and \(x=3\). Describe the key features of \(f(x)\).
Where \(f'>0\)
\(x<-1\) and \(x>3\): \(f\) is increasing
Where \(f'<0\)
\(-1
Where \(f'=0\)
\(x=-1\): local max; \(x=3\): local min
Where \(f'\) has minimum
\(x=1\): inflection point of \(f\)
Summary
Local max at \(x=-1\); local min at \(x=3\); inflection at \(x=1\)
Question 3
Sketch a possible function \(f(x)\) given that \(f'(x)=\sin x\) on \([0,2\pi]\), with \(f(0)=1\).
Integrate
\(f(x)=-\cos x+C\); \(f(0)=1\Rightarrow-1+C=1\Rightarrow C=2\)
Answer
\(f(x)=2-\cos x\): starts at \((0,1)\), max at \(x=\pi\) giving \(f(\pi)=3\), returns to \((2\pi,1)\)
Question 4
The graph of \(y=f(x)\) is a smooth curve with the following properties: decreasing on \((-\infty,2)\); increasing on \((2,\infty)\); concave up throughout. Sketch a possible \(y=f'(x)\).
Properties of \(f'\)
\(f'(x)<0\) for \(x<2\); \(f'(2)=0\); \(f'(x)>0\) for \(x>2\); since \(f''>0\), \(f'\) is increasing throughout
Sketch
Increasing function crossing \(x\)-axis at \(x=2\) — e.g. a straight line or gentle upward curve through \((2,0)\)
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