Mean Value Theorem and the police A sune patrol officer

Suppose f and g are functions that are continuous on [a, b] and differentiable on (a, b), where \(g(a) \neq g(b)\). Then, there is a point c in (a, b) at which \(\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}\). This result is known as the Generalized (or Cauchy's) Mean Value Theorem. a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem. b. Suppose \(f(x)=x^{2}-1\), g(x) = 4x + 2, and [a, b] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.

Suppose \(f^{\prime}(x)<0<f^{\prime \prime}(x)\) for x < a and \(f^{\prime}(x)>0>f^{\prime \prime}(x)\) for x > a. Prove that f is not differentiable at x = a. (Hint: Assume f is differentiable at a, and apply the Mean Value Theorem to \(f^{\prime}\).) More generally, show that if \(f^{\prime}\) and \(f^{\prime \prime}\) change sign at the same point, then f is not differentiable at that point.

The Jamaican sprinter Usain Bolt set a world record of 9.58 s in the 100-meter dash in the summer of 2009. Did his speed ever exceed 37 km/hr during the race? Explain.

Verify that the functions \(f(x)=\sin ^{2} x\) and \(g(x)=-\cos ^{2} x\) have the same derivative. What can you say about the difference f - g? Explain.

Verify that the functions \(f(x)=\tan ^{2} x\) and \(g(x)=\sec ^{2} x\) have the same derivative. What can you say about the difference f - g? Explain.

a. Show that the point o guaranteed to exist by the Mean Value Theorem for \(f(x)=x^{2}\) on [a, b] is the arithmetic mean of a and b; that is, c = (a + b) / 2. b. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a, b], where 0 < a < b, is the geometric mean of a and b; that is, \(c=\sqrt{a b}\).

Consider the quadratic function \(f(x)=A x^{2}+B x+C\), where A, B, and C are real numbers with \(A \neq 0\). Show that when the Mean Value Theorem is applied to f on the interval [a, b], the number c guaranteed by the theorem is the midpoint of the interval.

Interpret the Mean Value Theorem when it is applied to any linear function.

Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runners speed is 0 at the finish line.

Compare carefully to Exercise 29. A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another officer 30 mi along the highway. When the car reached the location of the second officer 30 min later, it was clocked going 60 mi/hr. Can the patrol officer conclude that the driver exceeded the speed limit?

A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the 60-mi/hr speed limit. Why can the officer conclude that the driver exceeded the speed limit?

Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. If the temperature gradient is large, it may lead to a weak layer of snow in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the following rule of thumb: If dT/dh exceeds \(10^{\circ} \mathrm{C}\)/m anywhere in the snowpack, conditions are favorable for weak layer formation and the risk of avalanche increases. Assume the temperature function is continuous and differentiable. a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0) the temperature is \(-12^{\circ} \mathrm{C}\). At a depth of 1.1 m, the temperature is \(2^{\circ} \mathrm{C}\). Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely? b. One mile away, a skier finds that the temperature at a depth of 1.4 m is \(-1^{\circ} \mathrm{C}\), and at the surface it is \(-12^{\circ} \mathrm{C}\). What can be concluded about the temperature gradient? Is the formation of a weak layer in her location likely? c. Because snow is an excellent insulator, the temperature of snow-covered ground is near \(0^{\circ} \mathrm{C}\). Furthermore, the surface temperature of snow in a particular area does not vary much from one location to the next. Explain why a weak layer is more likely to form in places where the snowpack is not too deep. d. The term isothermal is used to describe the situation where all layers of the snowpack are at the same temperature (typically near the freezing point). Is a weak layer likely to form in isothermal snow? Explain.

By visual inspection, locate all points on the graph at which the slope of the tangent line equals the average rate of change of the function over the interval [ -4,4].

Find all functions f whose derivative is \(f^{\prime}(x)=x+1\).

Without evaluating derivatives, which of the functions \(g(x)=2 x^{10}\), \(h(x)=x^{10}+2\), or \(p(x)=x^{10}-\ln 2\) have the same derivative as \(f(x)=x^{10}\)?

Without evaluating derivatives, which of the following functions have the same derivative: f(x) = In x, g(x) = In 2x, \(h(x)=\ln x^{2}\), \(p(x)=\ln 10 x^{2}\)?

Determine whether the following statements are true and give an explanation or counterexample. a. The continuous function f(x) = 1 - |x| satisfies the conditionsof the Mean Value Theorem on the interval [-1,1]. b. Two differentiable functions that differ by a constant always have the same derivative. c. If \(f^{\prime}(x)=0\), then f(x) = 10.

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. f(x) = x / (x + 2); [-1, 2]

At what points c does the conclusion of the Mean Value Theorem hold for \(f(x)=x^{3}\) on the interval [-10,10]?

Draw the graph of a function for which the conclusion of the Mean Value Theorem does not hold.

Explain the Mean Value Theorem with a sketch.

Explain why Rolle's Theorem cannot be applied to the function f(x) = |x| on the interval [-a, a] for any a > 0.

Draw the graph of a function for which the conclusion of Rolle's Theorem does not hold.

Explain Rolle's Theorem with a sketch.

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. \(f(x)=x(x-1)^{2} ;[0,1]\)

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. \(f(x)=\sin 2 x ;[0, \pi / 2]\)

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. \(f(x)=\cos 4 x ;[\pi / 8,3 \pi / 8]\)

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. f(x) = 1 - |x|; [-1, 1]

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. \(f(x)=1-x^{2 / 3} ;[-1,1]\)

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. \(f(x)=x^{3}-2 x^{2}-8 x ;[-2,4]\)

Concurrent measurements indicate that at an elevation of 6.1 km, the temperature is \(-10.3^{\circ} \mathrm{C}\), and at an elevation of 3.2 km, the temperature is \(8.0^{\circ} \mathrm{C}\). Based on the Mean Value Theorem, can you conclude that the lapse rate exceeds the threshold value of \(7^{\circ} \mathrm{C} / \mathrm{km}\) at some intermediate elevation? Explain.

The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 s (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag race is at least ___________ mi/hr/s.

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. \(f(x)=7-x^{2} ;[-1,2]\)

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. \(f(x)=3 \sin 2 x ;[0, \pi / 4]\)

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. \(f(x)=e^{x} ;[0, \ln 4]\)

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. f(x) = ln 2x ; [1, e]

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. \(f(x)=\sin ^{-1} x ;[0,1 / 2]\)

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. f(x) = x + 1 / x; [1,3]

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a, f(a)) and (b, f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. \(f(x)=2 x^{1 / 3} ;[-8,8]\)