21 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!
1 Where does it rise? ★★★
Let \(f(x)=x^2-6x+5\). Find \(f^{\prime}\), the critical point, and the intervals where \(f\) is increasing or decreasing. Identify the extremum.
2 True or false? ★★★
Decide and justify. (a) If \(f^{\prime}(x)>0\) on \((a,b)\), then \(f\) is increasing there. (b) If \(f^{\prime}(2)=0\), then \(f\) has a local extremum at \(x=2\).
3 A cubic hill and valley ★★★
Let \(g(x)=x^3-12x\). Find the critical points and classify them with the first derivative test.
4 Concavity check ★★★
For \(f(x)=x^3-9x^2+2\), find where the graph is concave up, concave down, and its inflection point.
5 Estimating a square ★★★
Use the linear approximation of \(f(x)=x^2\) at \(a=10\) to estimate \(10.2^2\). How far is the estimate from the exact value?
6 A hard-looking limit ★★★
Compute \(\displaystyle\lim_{x\to3}\dfrac{x^2-9}{x-3}\) using L’Hôpital’s rule, and check by factoring.
7 A growing square ★★★
The side \(s\) of a square tile grows at 3 cm/s. How fast is its area increasing when \(s=10\) cm?
8 The open box ★★★
A baker cuts four equal squares of side \(x\) inches from the corners of a 12 in by 12 in sheet of cardboard and folds up the sides to make an open box. Find \(x\) that maximizes the volume, and that volume.
9 Bending quartic ★★★
Let \(f(x)=x^4-6x^2\). Find the intervals of concavity and the inflection points.
10 Mean Value Theorem for a cube ★★★
Show that \(f(x)=x^3\) satisfies the hypotheses of the Mean Value Theorem on \([0,3]\) and find all values \(c\) it guarantees.
11 The turnpike driver ★★★
A driver enters a highway at mile marker 20 at 1:00 pm and exits at mile marker 170 at 3:00 pm. Show that the car traveled at exactly 75 mph at some moment. (Assume position is a differentiable function of time.)
12 Twice L’Hôpital ★★★
Compute \(\displaystyle\lim_{x\to0}\dfrac{e^{2x}-1-2x}{x^2}\).
13 The sliding ladder ★★★
A 10 ft ladder leans on a wall. Its foot slides away at 1.5 ft/s. How fast is the top moving down when the foot is 8 ft from the wall? What is the sign of the answer?
14 A cube root estimate ★★★
Use a linear approximation of \(f(x)=\sqrt[3]{x}\) at \(a=27\) to estimate \(\sqrt[3]{28}\).
15 The quartic that fooled the test ★★★
Study \(f(x)=x^4-4x^3\): critical points, extrema, concavity, and inflection points. Why does \(f^{\prime\prime}(0)=0\) not decide anything about \(x=0\)?
16 Closest point on a curve ★★★
Find the point on the curve \(y=\sqrt x\) that is closest to \((4,0)\), and the minimal distance.
17 The cheapest can ★★★
A closed cylindrical can must hold \(500\text{ cm}^3\). Find the radius and height that minimize the metal used (surface area), to the nearest hundredth.
18 Filling a cone ★★★
Water pours at 4 ft\(^3\)/min into a conical tank with the point down, radius 3 ft and height 6 ft at the top. How fast is the water level rising when the depth is 4 ft?
19 Two indeterminate forms ★★★
Compute (a) \(\displaystyle\lim_{x\to\infty}x^2e^{-x}\) and (b) \(\displaystyle\lim_{x\to0^+}x\ln x\). (Rewrite each as a quotient first.)
20 An inequality from the MVT ★★★
Use the Mean Value Theorem on \(f(t)=\ln(1+t)\) over \([0,x]\), with \(x>0\), to prove that \(\ln(1+x)
21 Find the cubic ★★★
The function \(f(x)=x^3+ax^2+bx\) has a local maximum at \(x=-1\) and a local minimum at \(x=3\). Find \(a\) and \(b\), the two extreme values, and the inflection point.
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