Applications of Derivatives: math test solutions, College – download the PDF

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Test solutions with the detailed point scale. Add up your points and spot what to review.

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

Sommaire

(a) \(f^{\prime}(x)=6x^2-6x-12=6(x-2)(x+1)\), so \(x=-1\) and \(x=2\). (1 pt)

(b) \(f^{\prime}>0\) on \((-\infty,-1)\) and \((2,\infty)\): increasing; \(f^{\prime}<0\) on \((-1,2)\): decreasing. (1.5 pts)

(a) \(f^{\prime\prime}(x)=12x-6\): concave down for \(x<\tfrac12\), concave up for \(x>\tfrac12\). (1 pt)

(b) \(f(\tfrac12)=\tfrac28-\tfrac34-6+4=-2.5\): the inflection point is \((0.5,-2.5)\). (1 pt)

\(f\) is continuous on \([1,9]\) and differentiable on \((1,9)\). (1 pt)

Average rate: \(\dfrac{3-1}{9-1}=\dfrac14\). (1 pt)

\(f^{\prime}(c)=\dfrac{1}{2\sqrt c}=\dfrac14\) gives \(\sqrt c=2\), so \(c=4\in(1,9)\). (1 pt)

(a) The fencing is \(2L+3w=240\), so \(L=120-1.5w\). (1 pt)

(b) \(A(w)=w(120-1.5w)\) for \(0

Then \(L=60\) m and \(A=2400\text{ m}^2\); \(A\) is \(0\) at both ends, so this is the maximum. (1.5 pts)

\(A=\pi r^2\), so \(\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\). (1 pt)

\(\dfrac{dA}{dt}=2\pi(15)(0.5)=15\pi\) ft\(^2\)/min. (1 pt)

\(15\pi\approx47.12\) ft\(^2\)/min. (1 pt)

(a) The form is \(\dfrac00\). By L’Hôpital: \(\dfrac{3e^{3x}}{2\cos2x}\to\dfrac32\). (1.5 pts)

(b) \(f(0)=0\), \(f^{\prime}(0)=1\), so \(L(x)=x\) and \(\ln1.05\approx0.05\) (the exact value is about \(0.0488\)). (1.5 pts)

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