1) Consider two runners \(v_1(t)\) and \(v_2(t).\) running at variable speeds. The runners start and end the race at the same time. Explain why both runners must be moving at the same speed at some point.

2) Two climbers start their climb from base camp and take two different routes, one steeper than the other, to reach the summit at the same time. At some point, both climbers are increasing in altitude at the same rate. Is this necessarily the case?

**answer**- Yes. It is implicit in the mean value theorem of integrals.

3) To enter a toll road, the driver must carry a card indicating the mileage entry point. The card is also time stamped. When preparing to pay the toll at the exit, the driver was surprised to receive a speeding ticket and the toll. Explain how this happened.

4) Let \(\displaystyle F(x)=∫^x _1(1−t)\,dt.\) find \( F′(2)\) and \(F'\) in \ ([1, 2].\)

**answer**- The average of \(F′(2)=−1;\) \(F'\) on \( [1,2]\) is \(−1/2.\)

**In Exercises 5–16, find each derivative using the Fundamental Theorem of Calculus, Part 1.**

5) \(\displaystyle \frac{d}{dx}\left[∫^x_1e^{−t^2}\,dt \right] \)

6) \(\displaystyle \frac{d}{dx}\left[∫^x_1e^{coste}\,dt \right] \)

**answer**- \(\displaystyle \frac{d}{dx}\left[∫^x_1e^{coste}\,dt \right] = e^{\cost}\)

7) \(\displaystyle \frac{d}{dx}\left[∫^x_3\sqrt{9−y^2}\,dy \right] \)

8) \(\displaystyle \frac{d}{dx}\left[∫^x_3\frac{ds}{\sqrt{16−s^2}} \right] \)

**answer**- \(\displaystyle \frac{d}{dx}\left[∫^x_3\frac{ds}{\sqrt{16−s^2}} \right] = \frac{1}{\sqrt{16−x ^2}}\)

9) \(\displaystyle \frac{d}{dx}\left[∫^{2x}x_t\,dt \right] \)

10) \(\displaystyle \frac{d}{dx}\left[∫^{\sqrt{x}}_0t\,dt \right] \)

**answer**- \(\displaystyle \frac{d}{dx}\left[∫^{\sqrt{x}}_0t\,dt \right] = \sqrt{x}\frac{d}{dx}\big(\sqrt {x}\grande)=\frac{1}{2}\)

11) \(\displaystyle \frac{d}{dx}\left[∫^{\sin x}_0\sqrt{1−t^2}\,dt \right] \)

12) \(\displaystyle \frac{d}{dx}\left[∫^1_{\cos x}\sqrt{1−t^2}\,dt \right] \)

**answer**- \(\displaystyle \frac{d}{dx}\left[∫^1_{\cos x}\sqrt{1−t^2}\,dt \right] = −\sqrt{1−\cos^2x} \frac{d}{dx}\grande(\cos x\grande)=|\sin x|\sin x\)

13) \(\displaystyle \frac{d}{dx}\left[∫^{\sqrt{x}}_1\frac{t^2}{1+t^4}\,dt \right] \)

14) \(\displaystyle \frac{d}{dx}\left[∫^{x^2}_1\frac{\sqrt{t}}{1+t}\,dt \right] \)

**answer**- \(\displaystyle \frac{d}{dx}\left[∫^{x^2}_1\frac{\sqrt{t}}{1+t}\,dt \right] = 2x\dfrac{|x |}{1+x^2}\)

15) \(\displaystyle \frac{d}{dx}\left[∫^{\ln x}_0e^t\,dt \right] \)

16) \(\displaystyle \frac{d}{dx}\left[∫^{e^{x}}_1\ln u^2\,du \right] \)

**answer**- \(\displaystyle \frac{d}{dx} \left[ ∫^{e^{x}}_1\ln u^2\,du \right] = \ln(e^{2x})\dfrac{d {dx}\grande(e^x\grande)=2xe^x\)

17) This is the graph of \(\displaystyle y=∫^x_0f(t)\,dt\), where \(f\) is a piecewise constant function.

arrive. In what interval is \(f\) positive? During what time interval is it negative? At what interval (if any) is it equal to zero?

b. What are the maximum and minimum values of \(f\)?

C. What is the average of \(f\)?

18) This is the graph of \(\displaystyle y=∫^x_0f(t)\,dt\), where \(f\) is a piecewise constant function.

arrive. In what interval is \(f\) positive? During what time interval is it negative? At what interval (if any) is it equal to zero?

b. What are the maximum and minimum values of \(f\)?

C. What is the average of \(f\)?

**answer**- arrive. \(f\) is positive on \([1,2]\) and \([5,6]\, and is negative on \([0,1]\) and \([3,4]\ , which is zero on \([2,3]\) and \([4,5]\).

b. The maximum value is \(2\), and the minimum value is \(−3\).

C. The average value is \(0\).

19) This is the graph of \(\displaystyle y=∫^x_0ℓ(t)\,dt\), where \(ℓ\) is a piecewise linear function.

arrive. Over what interval is \(ℓ\) positive? During what time interval is it negative? Which one, if any, is zero?

b. At what interval does \(ℓ\) grow? What is decreasing? If so, is it a constant?

C. What is the average of \(ℓ\)?

20) This is the graph of \(\displaystyle y=∫^x_0ℓ(t)\,dt\), where \( ℓ\) is a piecewise linear function.

arrive. Over what interval is \(ℓ\) positive? During what time interval is it negative? Which one, if any, is zero?

b. At what interval does \(ℓ\) grow? What is decreasing? Over what time interval, if any, is it constant?

C. What is the average of \(ℓ\)?

**answer**- arrive. \(ℓ\) is positive on \([0,1]\) and \([3,6]\) and negative on \([1,3]\).

b. It increases on \([0,1]\) and \([3,5]\), and remains unchanged on \([1,3]\) and \([5,6]\) Change.

C. Its mean is \(\dfrac{1}{3}\).

**In exercises 21-26, use a calculator to estimate the area under the curve by calculating \(T_{10}\), that is, using the average of the Riemann sums at the left and right ends of \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.**

21) [T] \( y=x^2\) in \( [0,4]\)

22) [T] \(y=x^3+6x^2+x−5\) in \( [−4,2]\)

**answer**- \(\displaystyle T_{10}=49.08,\quad ∫^3_{−2}\left(x^3+6x^2+x−5\right)\,dx=48\)

23) [T] \(y=\sqrt{x^3}\) in \( [0,6]\)

24) [T] \(y=\sqrt{x}+x^2\) in \([1,9]\)

**answer**- \(\displaystyle T_{10}=260.836,\quad ∫^9_1(\sqrt{x}+x^2)\,dx=260\)

25) [T] \(\displaystyle ∫(\cos x−\sin x)\,dx\) sobre \(\displaystyle [0,π]\)

26) [T] \(\displaystyle ∫\frac{4}{x^2}\,dx\) 清醒 \(\displaystyle [1,4]\)

**answer**- \(\displaystyle T_{10}=3.058,\quad ∫^4_1\frac{4}{x^2}\,dx=3\)

**In Exercises 27–46, compute each definite integral using the Fundamental Theorem of Calculus, Part 2.**

27) \(\displaystyle ∫^2_{−1}(x^2−3x)\,dx\)

28) \(\displaystyle ∫^3_{−2}(x^2+3x−5)\,dx\)

**answer**- \(\displaystyle F(x)=\frac{x^3}{3}+\frac{3x^2}{2}−5x,\quad F(3)−F(−2)=−\frac{ 35}{6}\)

29) \(\displaystyle ∫^3_{−2}(t+2)(t−3)dt\)

30) \(\displaystyle ∫^3_2(t^2−9)(4−t^2)dt\)

**answer**- \(F(x)=−\dfrac{t^5}{5}+\dfrac{13t^3}{3}−36t,\quad F(3)−F(2)=\frac{62}{ 15}\)

31) \(\displaystyle ∫^2_1x^9\,dx\)

32) \(\displaystyle ∫^1_0x^{99}\,dx\)

**answer**- \(F(x)=\dfrac{x^{100}}{100},\quad F(1)−F(0)=\dfrac{1}{100}\)

33) \(\displaystyle ∫^8_4(4t^{5/2}−3t^{3/2})dt\)

34) \(\displaystyle ∫^4_{1/4}\left(x^2−\dfrac{1}{x^2}\right)\,dx\)

**answer**- \(F(x)=\dfrac{x^3}{3}+\dfrac{1}{x},\quad F(4)−F\left(\tfrac{1}{4}\right)= \frac{1125}{64}\)

35) \(\displaystyle ∫^2_1\frac{2}{x^3}\,dx\)

36) \(\displaystyle ∫^4_1\frac{1}{2\sqrt{x}}\,dx\)

**answer**- \(F(x)=\sqrt{x},\quad F(4)−F(1)=1\)

37) \(\displaystyle ∫^4_1\frac{2−\sqrt{t}}{t^2}\,dt\)

38) \(\displaystyle ∫^{16}_1\frac{dt}{t^{1/4}}\)

**answer**- \( F(t)=\frac{4}{3}t^{3/4},\quad F(16)−F(1)=\frac{28}{3}\)

39) \(\displaystyle ∫^{2π}_0\cos θ\,dθ\)

40) \(\displaystyle ∫^{π/2}_0\sen θ\,dθ\)

**answer**- \(F(θ)=−\cos θ,\quad F(\frac{π}{2})−F(0)=1\)

41) \(\displaystyle ∫^{π/4}_0\sec^2 θ\,dθ\)

42) \(\displaystyle ∫^{π/4}_0\sec θ\tan θ\,dθ\)

**answer**- \(F(θ)=\sec θ,\quad F(\frac{π}{4})−F(0)=\sqrt{2}−1\)

43) \(\displaystyle ∫^{π/4}_{π/3}\csc θ\cot θ\,dθ\)

44) \(\displaystyle ∫^{π/2}_{π/4}\csc^2 θ\,dθ\)

**answer**- \(F(θ)=−\cot θ,\quad F(\frac{π}{2})−F(\frac{π}{4})=1\)

45) \(\displaystyle ∫^2_1\left(\frac{1}{t^2}−\frac{1}{t^3}\right)\,dt\)

46) \(\displaystyle ∫^{−1}_{−2}\left(\frac{1}{t^2}−\frac{1}{t^3}\right)\,dt\)

**answer**- \( F(t)=−\dfrac{1}{t}+\dfrac{1}{2t^2},\quad F(−1)−F(−2)=\frac{7}{8} \)

**In Exercises 47 - 50, use the evaluation theorem to express integrals as functions \(F(x).\)**

47) \(\displaystyle ∫^x_at^2\,dt\)

48) \(\displaystyle ∫^x_1e^t\,dt\)

**answer**- \(F(x)=e^x−e\)

49) \(\displaystyle ∫^x_0\cost\,dt\)

50) \(\displaystyle ∫^x_{−x}\sin t\,dt\)

**answer**- \(F(x)=0\)

**In Exercises 51 - 54, the roots of the integrand are determined to eliminate absolute values, then computed using the Fundamental Theorem of Calculus (Part 2).**

51) \(\displaystyle ∫^3_{−2}|x|\,dx\)

52) \(\displaystyle ∫^4_{−2}∠t^2−2t−3∠\,dt\)

**answer**- \(\displaystyle ∫^{−1}_{−2}(t^2−2t−3)\,dt−∫^3_{−1}(t^2−2t−3)\,dt+∫^4_3 (t^2−2t−3)\,dt=\frac{46}{3}\)

53) \(\displaystyle ∫^π_0|\cost|\,dt\)

54) \(\displaystyle ∫^{π/2}_{−π/2}|\sin t|\,dt\)

**answer**- \(\displaystyle −∫^0_{−π/2}\sen t\,dt+∫^{π/2}_0\sin t\,dt=2\)

55) Assume that the sunshine hours of a certain day in Seattle are represented by the function \(−3.75\cos(\frac{πt}{6})+12.25\), where \(t\) is given in months, and\( t=0\) corresponds to the winter solstice.

arrive. What is the average number of hours of sunshine in a year?

b. At what moment \(t_1\) and \(t_2\), where \(0≤t_1

C. Write an integral representing the total sunshine hours in Seattle between \(t_1\) and \(t_2\)

d. Calculate the average sunshine hours in Seattle between \(t_1\) and \(t_2\), where \(0≤t_1

56) Assume that the gasoline consumption rate in the United States can be represented by a sine function of the form \((11.21−\cos(\frac{πt}{6}))×10^9\) gal/ mine.

arrive. What is the average monthly consumption? For what value of \(t\) is the rate at time \(t\) equal to the average rate?

b. How many gallons of gasoline does the United States consume in a year?

C. Write an integral representing the average monthly natural gas consumption in the United States for the year between \((t=3)\) at the beginning of April and \((t=9).\) at the end of September

**answer**- arrive. The mean is \(11.21×10^9\), since \(\cos(\frac{πt}{6})\) has a period of 12 and an integral of 0 in any period. When \(\cos(\frac{πt}{6})=0\), \(t=3\) and \(t=9\), the consumption is equal to the average.

b. The total consumption is the average rate multiplied by the duration: \(11.21×12×10^9=1.35×10^{11}\)

C. \(\displaystyle 10^9(11,21−\frac{1}{6}∫^9_3\cos(\frac{πt}{6})\,dt)=10^9(11,21+ 2π)=11,84x10^9\)

57) Explain why, if \(f\) is continuous on \([a,b],\), then there is at least one point\( c∈[a,b]\) such that \(\displaystyle f(c ) =\frac{1}{b−a}∫^b_af(t)\,dt.\)

58) Explain why, if \(f\) is continuous on \([a,b]\) and not equal to a constant, then there exists at least one point\( M∈[a,b]\) such that\(\displaystyle f (M)=\frac{1}{b−a}∫^b_af(t)\,dt\) and at least one point\(m∈[a,b]\) such that\( \displaystyle f(m)< \frac{1}{b−a}∫^b_af(t)\,dt\).

**answer**- If \(f\) is not constant, its mean value is strictly less than the maximum value and greater than the minimum value, which is obtained on \([a,b]\) by the extreme value theorem.

59) Kepler's first law states that planets move in elliptical orbits with the sun at one focus. The closest point in a planet's orbit to the sun is called**perihelion**(for Earth, it currently occurs around January 3) and the farthest point is called**aphelion**(For Earth, it currently occurs around July 4th.) Kepler's second law states that planets sweep equal areas in their elliptical orbits in equal times. Therefore, the two arcs shown in the diagram below sweep in equal time. What time of year is Earth moving fastest in its orbit? When does it move the slowest? Kepler's first law states that planets move in elliptical orbits with the sun at one focus. The closest point in a planet's orbit to the Sun is called perihelion (for Earth, currently occurs around January 3), and the furthest point is called aphelion (for Earth, currently occurs around July 4). Kepler's second law states that planets sweep out equal areas in their elliptical orbits in equal times. Therefore, the two arcs shown in the diagram below sweep in equal time. What time of year is Earth moving fastest in its orbit? When does it move the slowest?

60) The coordinates of a point on the ellipse whose major axis length is\(2a\) and whose minor axis length is\(2b\) are\((a\cos θ,b\sin θ),\quad 0≤θ≤2π .\)

arrive. Prove that the distance from the point to the focal point at \((−c,0)\) is\( d(θ)=a+c\cos θ\), where\(c=\sqrt{a^2− b^ 2}\).

Using these coordinates shows that the average distance \( \bar{d}\) from a point on the ellipse to the focus at \((−c,0),\) is \(to\) with respect to the angle \(θ\) .

**answer**- A。 \( d^2θ=(a\cos θ+c)^2+b^2\sin^2 θ=a^2+c^2\cos^2 θ+2ac\cos θ=(a+c \cos yo)^2;\)

b. \(\displaystyle \bar{d}=\frac{1}{2π}∫^{2π}_0(a+2c\cos θ)\,dθ=a\)

61) As mentioned above, according to Kepler's law, the earth's orbit is an ellipse with the sun at one focus. The perihelion of the Earth's orbit around the sun is 147,098,290 km and the aphelion is 152,098,232 km.

arrive. Place the principal axis along the \(x\) axis and find the average distance from the Earth to the Sun.

b. The classic definition of an astronomical unit (AU) is the distance from the Earth to the sun, calculated as the average of the perihelion and aphelion distances. Is this definition reasonable?

62) The gravitational force between the sun and the planet is \(F(θ)=\dfrac{GmM}{r^2(θ)}\), where \(m\) is the mass of the planet, and \(M\) is The mass of the sun, \(G\) is a universal constant, \(r(θ)\) is the angle between the planet and the planet\(θ\ ) and the major axis of its orbit. Assuming \(M\), \(m\) and elliptic parameters \(a\) and \(b\) (average lengths of the major and minor axes) are given, establish but do not evaluate - an integral representation according to \(G,\,m,\,M,\,a,\,b\) Average gravitational force between the sun and the planet.

**answer**- Average gravity\(\displaystyle = \frac{GmM}{2}∫^{2π}_0\frac{1}{(a+2\sqrt{a^2−b^2}\cos θ)^2} \ ,dθ\).
The displacement of the rest of the mass connected to the spring satisfies the simple harmonic motion equation \( x(t)=Acos(ωt−ϕ),\) where \(ϕ\) is the phase constant, and \(ω \) is the angular frequency , \(A\) is the amplitude. Find the average velocity, average velocity (magnitude of velocity), average displacement, and average distance from rest (magnitude of displacement) of the mass.