Prove that d dx vx ux ft dt fvxvx fuxux

Graphical Reasoning In Exercises 1– 4, use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. \(\int_{0}^{\pi} \frac{4}{x^{2}+1} d x\) Text Transcription: \int_{0}^{\pi} \frac{4}{x^{2}+1} d x

Graphical Reasoning In Exercises 1– 4, use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. \(\int_{0}^{\pi} \cos x d x\) Text Transcription: \int_{0}^{\pi} \cos x d x

Graphical Reasoning In Exercises 1– 4, use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. \(\int_{-2}^{2} x \sqrt{x^{2}+1} d x\) Text Transcription: \int_{-2}^{2} x \sqrt{x^{2}+1} d x

Graphical Reasoning In Exercises 1– 4, use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. \(\int_{-2}^{2} x \sqrt{2-x} d x\) Text Transcription: \int_{-2}^{2} x \sqrt{2-x} d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{2} 6 x d x\) Text Transcription: \int_{0}^{2} 6 x d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{4}^{9} 5 d v\) Text Transcription: \int_{4}^{9} 5 d v

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-1}^{0}(2 x-1) d x\) Text Transcription: \int_{-1}^{0}(2 x-1) d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{2}^{5}(-3 v+4) d v\) Text Transcription: \int_{2}^{5}(-3 v+4) d v

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-1}^{1}\left(t^{2}-2\right) d t\) Text Transcription: \int_{-1}^{1}\left(t^{2}-2\right) d t

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{1}^{7}\left(6 x^{2}+2 x-3\right) d x\) Text Transcription: \int_{1}^{7}\left(6 x^{2}+2 x-3\right) d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{1}(2 t-1)^{2} d t\) Text Transcription: \int_{0}^{1}(2 t-1)^{2} d t

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-1}^{1}\left(t^{3}-9 t\right) d t\) Text Transcription: \int_{-1}^{1}\left(t^{3}-9 t\right) d t

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{1}^{2}\left(\frac{3}{x^{2}}-1\right) d x\) Text Transcription: \int_{1}^{2}\left(\frac{3}{x^{2}}-1\right) d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-2}^{-1}\left(u-\frac{1}{u^{2}}\right) d u\) Text Transcription: \int_{-2}^{-1}\left(u-\frac{1}{u^{2}}\right) d u

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{1}^{4} \frac{u-2}{\sqrt{u}} d u\) Text Transcription: \int_{1}^{4} \frac{u-2}{\sqrt{u}} d u

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-3}^{3} v^{1 / 3} d v\) Text Transcription: \int_{-3}^{3} v^{1 / 3} d v

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-1}^{1}(\sqrt[3]{t}-2) d t) Text Transcription: \int_{-1}^{1}(\sqrt[3]{t}-2) d t

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{1}^{8} \sqrt{\frac{2}{x}} d x\) Text Transcription: \int_{1}^{8} \sqrt{\frac{2}{x}} d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{1} \frac{x-\sqrt{x}}{3} d x\) Text Transcription: \int_{0}^{1} \frac{x-\sqrt{x}}{3} d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{2}(2-t) \sqrt{t} d t\) Text Transcription: \int_{0}^{2}(2-t) \sqrt{t} d t

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-1}^{0}\left(t^{1 / 3}-t^{2 / 3}\right) d t\) Text Transcription: \int_{-1}^{0}\left(t^{1 / 3}-t^{2 / 3}\right) d t

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{-8}^{-1} \frac{x-x^{2}}{2 \sqrt[3]{x}} d x\) Text Transcription: \int_{-8}^{-1} \frac{x-x^{2}}{2 \sqrt[3]{x}} d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{5}|2 x-5| d x\) Text Transcription: \int_{0}^{5}|2 x-5| d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{1}^{4}(3-|x-3|) d x\) Text Transcription: \int_{1}^{4}(3-|x-3|) d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{4}\left|x^{2}-9\right| d x\) Text Transcription: \int_{0}^{4}\left|x^{2}-9\right| d x

In Exercises 5–26, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. \(\int_{0}^{4}\left|x^{2}-4 x+3\right| d x\) Text Transcription: \int_{0}^{4}\left|x^{2}-4 x+3\right| d x

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{0}^{\pi}(1+\sin x) d x\) Text Transcription: \int_{0}^{\pi}(1+\sin x) d x

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{0}^{\pi}(2+\cos x) d x\) Text Transcription: \int_{0}^{\pi}(2+\cos x) d x

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{0}^{\pi / 4} \frac{1-\sin ^{2} \theta}{\cos ^{2} \theta} d \theta\) Text Transcription: \int_{0}^{\pi / 4} \frac{1-\sin ^{2} \theta}{\cos ^{2} \theta} d \theta

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{0}^{\pi / 4} \frac{\sec ^{2} \theta}{\tan ^{2} \theta+1} d \theta\) Text Transcription: \int_{0}^{\pi / 4} \frac{\sec ^{2} \theta}{\tan ^{2} \theta+1} d \theta

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{-\pi / 6}^{\pi / 6} \sec ^{2} x d x\) Text Transcription: \int_{-\pi / 6}^{\pi / 6} \sec ^{2} x d x

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{\pi / 4}^{\pi / 2}\left(2-\csc ^{2} x\right) d x\) Text Transcription: \int_{\pi / 4}^{\pi / 2}\left(2-\csc ^{2} x\right) d x

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{-\pi / 3}^{\pi / 3} 4 \sec \theta \tan \theta d \theta\) Text Transcription: \int_{-\pi / 3}^{\pi / 3} 4 \sec \theta \tan \theta d \theta

In Exercises 27–34, evaluate the definite integral of the trigonometric function. Use a graphing utility to verify your result. \(\int_{-\pi / 2}^{\pi / 2}(2 t+\cos t) d t\) Text Transcription: \int_{-\pi / 2}^{\pi / 2}(2 t+\cos t) d t

In Exercises 35–38, determine the area of the given region. \(y=x-x^{2}\) Text Transcription: y=x-x^{2}

In Exercises 35–38, determine the area of the given region. \(y=\frac{1}{x^{2}}\) Text Transcription: y=\frac{1}{x^{2}}

In Exercises 35–38, determine the area of the given region. y= cos x

In Exercises 35–38, determine the area of the given region. y=x + sin x

In Exercises 39– 44, find the area of the region bounded by the graphs of the equations. \(y=5 x^{2}+2\), x=0, x=2, y=0 Text Transcription: y=5 x^{2}+2

In Exercises 39– 44, find the area of the region bounded by the graphs of the equations. \(y=x^{3}+x\), x=2, y=0 Text Transcription: y=x^{3}+x

In Exercises 39– 44, find the area of the region bounded by the graphs of the equations. \(y=1+\sqrt[3]{x}\), x=0, x=8, y=0 Text Transcription: y=1+\sqrt[3]{x}

In Exercises 39– 44, find the area of the region bounded by the graphs of the equations. \(y=(3-x) \sqrt{x}\), y=0 Text Transcription: y=(3-x) \sqrt{x}

In Exercises 39– 44, find the area of the region bounded by the graphs of the equations. \(y=-x^{2}+4 x\), y=0 Text Transcription: y=-x^{2}+4 x

In Exercises 39– 44, find the area of the region bounded by the graphs of the equations. \(y=1-x^{4}\), y=0 Text Transcription: y=1-x^{4}

In Exercises 45–50, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. \(f(x)=x^{3}\), [0,3] Text Transcription: f(x)=x^{3}

In Exercises 45–50, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. \(f(x)=\frac{9}{x^{3}}\), [1,3] Text Transcription: f(x)=\frac{9}{x^{3}}

In Exercises 45–50, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. \(f(x)=\sqrt{x}\), [4,9] Text Transcription: f(x)=\sqrt{x}

In Exercises 45–50, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. \(f(x)=x-2 \sqrt{x}\), [0,2] Text Transcription: f(x)=x-2 \sqrt{x}

In Exercises 45–50, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. \(f(x)=2 \sec ^{2} x,\) \([-\pi / 4, \pi / 4]\) Text Transcription: f(x)=2 \sec ^{2} x [-\pi / 4, \pi / 4]

In Exercises 45–50, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. \(f(x)=\cos x, \)\([-\pi / 3, \pi / 3]\) Text Transcription: f(x)=\cos x, [-\pi / 3, \pi / 3]

In Exercises 51–56, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \(f(x)=9-x^{2}\), [-3,3] Text Transcription: f(x)=9-x^{2}

In Exercises 51–56, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \(f(x)=\frac{4(x^{2}+1)}{x^{2}}\), [1,3] Text Transcription: f(x)=\frac{4(x^{2}+1)}{x^{2}}

In Exercises 51–56, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \(f(x)=x^{3}\), [0,1] Text Transcription: f(x)=x^{3}

In Exercises 51–56, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \(f(x)=4 x^{3}-3 x^{2}\), [-1,2] Text Transcription: f(x)=4 x^{3}-3 x^{2}

In Exercises 51–56, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. f(x) = sin x, [0,\(\pi\)] Text Transcription: pi

In Exercises 51–56, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. f(x)=cos x, \([0, \pi/2]\) Text Transcription: [0, \pi/2]

Velocity The graph shows the velocity, in feet per second, of a car accelerating from rest. Use the graph to estimate the distance the car travels in 8 seconds.

Velocity The graph shows the velocity, in feet per second, of a decelerating car after the driver applies the brakes. Use the graph to estimate how far the car travels before it comes to a stop.

The graph of f is shown in the figure. (a) Evaluate \(\int_{1}^{7} f(x) d x\) (b) Determine the average value of on the interval [1,7] (c) Determine the answers to parts (a) and (b) if the graph is translated two units upward Text Transcription: \int_{1}^{7} f(x) d x

If r’(t) represents the rate of growth of a dog in pounds per year, what does r(t) represent? What does \(\int_{2}^{6} r^{\prime}(t) d t\) represent about the dog? Text Transcription: \int_{2}^{6} r^{\prime}(t) d t

Force The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of sec x, where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is [0,\(\pi)/3] and F(0) = 500 (a) Find as a function of x (b) Find the average force exerted by the press over the interval [0,\(\pi)/3] Text Transcription: pi

Blood Flow The velocity v of the flow of blood at a distance r from the central axis of an artery of radius R is \(v=k\left(R^{2}-r^{2}\right)\) where k is the constant of proportionality. Find the average rate of flow of blood along a radius of the artery. (Use 0 and R as the limits of integration.) Text Transcription: v=k\left(R^{2}-r^{2}\right)

Respiratory Cycle The volume V in, liters, of air in the lungs during a five-second respiratory cycle is approximated by the model \(V=0.1729 t+0.1522 t^{2}-0.0374 t^{3}\) where t is the time in seconds. Approximate the average volume of air in the lungs during one cycle Text Transcription: V=0.1729 t+0.1522 t^{2}-0.0374 t^{3}

Average Sales A company fits a model to the monthly sales data for a seasonal product. The model is \(S(t)=\frac{t}{4}+1.8+0.5 \sin \left(\frac{\pi t}{6}\right)\), \(0 \leq t \leq 24\) where S is sales (in thousands) and t is time in months. (a) Use a graphing utility to graph \(f(t)=0.5 \sin (\pi t / 6)\) for 0<t<24. Use the graph to explain why the average value of f(t) is 0 over the interval. (b) Use a graphing utility to graph S(t) and the line g(t) = t/4 +1.8 in the same viewing window. Use the graph and the result of part (a) to explain why g is called the trend line. Text Transcription: f(t)=0.5 \sin (\pi t / 6) S(t)=\frac{t}{4}+1.8+0.5 \sin \left(\frac{\pi t}{6}\right) 0 \leq t \leq 24

Modeling Data An experimental vehicle is tested on a straight track. It starts from rest, and its velocity v (in meters per second) is recorded every 10 seconds for 1 minute (see table). (a) Use a graphing utility to find a model of the form \(v=a t^{3}+b t^{2}+c t+d\) for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the Fundamental Theorem of Calculus to approximate the distance traveled by the vehicle during the test. Text Transcription: v=a t^{3}+b t^{2}+c t+d

The graph of f is shown in the figure. The shaded region has an area of 1.5, and \(\int_{0}^{6} f(x) d x=3.5\) Use this information to fill in the blanks. (a) \(\int_{0}^{2} f(x) d x\) = ___ (b) \(\int_{2}^{6} f(x) d x=\) ___ (c) \(\int_{0}^{6}|f(x)| d x=\) ___ (d) \(\int_{0}^{2}-2 f(x) d x\) = ___ (e) \(\int_{0}^{6}[2+f(x)] d x\) = ___ (f) The average value of f over the interval [0,6] is ___ Text Transcription: \int_{0}^{6} f(x) d x=3.5 \int_{0}^{2} f(x) d x \int_{2}^{6} f(x) d x \int_{0}^{6}|f(x)| d x \int_{0}^{2}-2 f(x) d x \int_{0}^{6}[2+f(x)] d x

In Exercises 67–72, find F as a function of x and evaluate it at x=2, x=5 and x=8 \(F(x)=\int_{0}^{x}(4 t-7) d t\) Text Transcription: F(x)=\int_{0}^{x}(4 t-7) d t

In Exercises 67–72, find F as a function of x and evaluate it at x=2, x=5 and x=8 \( F(x)=\int_{2}^{x}\left(t^{3}+2 t-2\right) d t\) Text Transcription: F(x)=\int_{2}^{x}\left(t^{3}+2 t-2\right) d t

In Exercises 67–72, find F as a function of x and evaluate it at x=2, x=5 and x=8 \(F(x)=\int_{1}^{x} \frac{20}{v^{2}} d v\) Text Transcription: F(x)=\int_{1}^{x} \frac{20}{v^{2}} d v

In Exercises 67–72, find F as a function of x and evaluate it at x=2, x=5 and x=8 \(F(x)=\int_{2}^{x}-\frac{2}{t^{3}} d t\) Text Transcription: F(x)=\int_{2}^{x}-\frac{2}{t^{3}} d t

In Exercises 67–72, find F as a function of x and evaluate it at x=2, x=5 and x=8 \(F(x)=\int_{1}^{x} \cos \theta d \theta\) Text Transcription: F(x)=\int_{1}^{x} \cos \theta d \theta

In Exercises 67–72, find F as a function of x and evaluate it at x=2, x=5 and x=8 \(F(x)=\int_{0}^{x} \sin \theta d \theta\) Text Transcription: F(x)=\int_{0}^{x} \sin \theta d \theta

Let \(g(x)=\int_{0}^{x} f(t) d t\) where f is the function whose graph is shown in the figure. (a) Estimate and g(0), g(2), g(4), g(6), and g(8). (b) Find the largest open interval on which is increasing. Find the largest open interval on which g is decreasing. (c) Identify any extrema of g (d) Sketch a rough graph of g Text Transcription: g(x)=\int_{0}^{x} f(t) d t

Let \(g(x)=\int_{0}^{x} f(t) d t\) where f is the function whose graph is shown in the figure. (a) Estimate and g(0), g(2), g(4), g(6), and g(8). (b) Find the largest open interval on which is increasing. Find the largest open interval on which g is decreasing. (c) Identify any extrema of g (d) Sketch a rough graph of g Text Transcription: g(x)=\int_{0}^{x} f(t) d t

In Exercises 75– 80, (a) integrate to find F as a function of x and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). \(F(x)=\int_{0}^{x}(t+2) d t\) Text Transcription: F(x)=\int_{0}^{x}(t+2) d t

In Exercises 75– 80, (a) integrate to find F as a function of x and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). \(F(x)=\int_{0}^{x} t\left(t^{2}+1\right) d t\) Text Transcription: F(x)=\int_{0}^{x} t\left(t^{2}+1\right) d t

In Exercises 75– 80, (a) integrate to find F as a function of x and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). \(F(x)=\int_{8}^{x} \sqrt[3]{t} d t\) Text Transcription: F(x)=\int_{8}^{x} \sqrt[3]{t} d t

In Exercises 75– 80, (a) integrate to find F as a function of x and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). \(F(x)=\int_{4}^{x} \sqrt{t} d t\) Text Transcription: F(x)=\int_{4}^{x} \sqrt{t} d t

In Exercises 75– 80, (a) integrate to find F as a function of x and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). \(F(x)=\int_{\pi / 4}^{x} \sec ^{2} t d t\) Text Transcription: F(x)=\int_{\pi / 4}^{x} \sec ^{2} t d t

In Exercises 75– 80, (a) integrate to find F as a function of x and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). \(F(x)=\int_{\pi / 3}^{x} \sec t \tan t d t\) Text Transcription: F(x)=\int_{\pi / 3}^{x} \sec t \tan t d t

In Exercises 81–86, use the Second Fundamental Theorem of Calculus to find F’(x) \(F(x)=\int_{-2}^{x}\left(t^{2}-2 t\right) d t\) Text Transcription: F(x)=\int_{-2}^{x}\left(t^{2}-2 t\right) d t

In Exercises 81–86, use the Second Fundamental Theorem of Calculus to find F’(x) \(F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t\) Text Transcription: F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t

In Exercises 81–86, use the Second Fundamental Theorem of Calculus to find F’(x) \(F(x)=\int_{-1}^{x} \sqrt{t^{4}+1} d t\) Text Transcription: F(x)=\int_{-1}^{x} \sqrt{t^{4}+1} d t

In Exercises 81–86, use the Second Fundamental Theorem of Calculus to find F’(x) \(F(x)=\int_{1}^{x} \sqrt[4]{t} d t\) Text Transcription: F(x)=\int_{1}^{x} \sqrt[4]{t} d t

In Exercises 81–86, use the Second Fundamental Theorem of Calculus to find F’(x) \(F(x)=\int_{0}^{x} t \cos t d t\) Text Transcription: F(x)=\int_{0}^{x} t \cos t d t

In Exercises 81–86, use the Second Fundamental Theorem of Calculus to find F’(x) \(F(x)=\int_{0}^{x} \sec ^{3} t d t\) Text Transcription: F(x)=\int_{0}^{x} \sec ^{3} t d t

In Exercises 87–92, find F’(x) \(F(x)=\int_{x}^{x+2}(4 t+1) d t\) Text Transcription: F(x)=\int_{x}^{x+2}(4 t+1) d t

In Exercises 87–92, find F’(x) \(F(x)=\int_{-x}^{x} t^{3} d t\) Text Transcription: F(x)=\int_{-x}^{x} t^{3} d t

In Exercises 87–92, find F’(x) \(F(x)=\int_{0}^{\sin x} \sqrt{t} d t\) Text Transcription: F(x)=\int_{0}^{\sin x} \sqrt{t} d t

In Exercises 87–92, find F’(x) \(F(x)=\int_{2}^{x^{2}} \frac{1}{t^{3}} d t\) Text Transcription: F(x)=\int_{2}^{x^{2}} \frac{1}{t^{3}} d t

In Exercises 87–92, find F’(x) \(F(x)=\int_{0}^{x^{3}} \sin t^{2} d t\) Text Transcription: F(x)=\int_{0}^{x^{3}} \sin t^{2} d t

In Exercises 87–92, find F’(x) \(F(x)=\int_{0}^{x^{2}} \sin \theta^{2} d \theta\) Text Transcription: F(x)=\int_{0}^{x^{2}} \sin \theta^{2} d \theta

Graphical Analysis Sketch an approximate graph of g on the interval \(0 \leq x \leq 4\) where \(g(x)=\int_{0}^{u} f(t) d t\)Identify the x-coordinate of an extremum of g. To print an enlarged copy of the graph, go to the website www.mathgraphs.com Text Transcription: 0 \leq x \leq 4 g(x)=\int_{0}^{u} f(t) d t

Use the graph of the function f shown in the figure and the function g defined by \(g(x)=\int_{0}^{x} f(t) d t\) a) Complete the table. (b) Plot the points from the table in part (a) and graph g (c) Where does g have its minimum? Explain. (d) Where does g have a maximum? Explain. (e) On what interval does g increase at the greatest rate? Explain. (f) Identify the zeros of g Text Transcription: g(x)=\int_{0}^{x} f(t) d t

Cost The total cost (in dollars) of purchasing and maintaining a piece of equipment for years is \(C(x)=5000\left(25+3 \int_{0}^{x} t^{1 / 4} d t\right)\) (a) Perform the integration to write C as a function of x (b) Find C(1), C(5), and C(10) Text Transcription: C(x)=5000(25+3 \int_{0}^{x} t^{1 / 4} d t)

Area The area between the graph of the function \(g(t)=4-4 / t^{2}\) and the t-axis over the interval [1,x] is \(A(x)=\int_{1}^{x}\left(4-\frac{4}{t^{2}}\right) d t\) (a) Find the horizontal asymptote of the graph of g. (b) Integrate to find as a function of x. Does the graph of A have a horizontal asymptote? Explain. Text Transcription: g(t)=4-4 / t^{2} A(x)=\int_{1}^{x}\left(4-\frac{4}{t^{2}}\right) d t

In Exercises 97–102, the velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. v(t)=5 t-7\(0 \leq t \leq 3\) Text Transcription: 0 \leq t \leq 3

In Exercises 97–102, the velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \(v(t)=t^{2}-t-12\), \( 1 \leq t \leq 5\) Text Transcription: v(t)=t^{2}-t-12 1 \leq t \leq 5

In Exercises 97–102, the velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \(v(t)=t^{3}-10 t^{2}+27 t-18\), \(1 \leq t \leq 7\) Text Transcription: v(t)=t^{3}-10 t^{2}+27 t-18 1 \leq t \leq 7

In Exercises 97–102, the velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \(v(t)=t^{3}-8 t^{2}+15 t\), \(0 \leq t \leq 5\) Text Transcription: v(t)=t^{3}-8 t^{2}+15 t 0 \leq t \leq 5

In Exercises 97–102, the velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \(v(t)=\frac{1}{\sqrt{t}}\), \(1 \leq t \leq 4\) Text Transcription: v(t)=\frac{1}{\sqrt{t}} 1 \leq t \leq 4

In Exercises 97–102, the velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. v(t) = cos t, \(0 \leq t \leq 3 \pi\) Text Transcription: 0 \leq t \leq 3 \pi

A particle is moving along the x-axis. The position of the particle at time t is given by \(x(t)=t^{3}-6 t^{2}+9 t-2\), \(0 \leq t \leq 5\) Find the total distance the particle travels in 5 units of time. Text Transcription: x(t)=t^{3}-6 t^{2}+9 t-2 0 \leq t \leq 5

Repeat Exercise 103 for the position function given by \(x(t)=(t-1)(t-3)^{2}, 0 \leq t \leq 5 .\) Text Transcription: x(t)=(t-1)(t-3)^{2} 0 \leq t \leq 5 .

Water Flow Water flows from a storage tank at a rate of 500-5t liters per minute. Find the amount of water that flows out of the tank during the first 18 minutes.

Oil Leak At 1:00 P.M., oil begins leaking from a tank at a rate of 4+0.75t gallons per hour. (a) How much oil is lost from 1:00 P.M. to 4:00 P.M.? (b) How much oil is lost from 4:00 P.M. to 7:00 P.M.? (c) Compare your answers from parts (a) and (b). What do you notice?

In Exercises 107–110, describe why the statement is incorrect. \(\int_{-1}^{1} x^{-2} d x=[-x-1]_{-1}^{1}=(-1)-1=-2\) Text Transcription: \int_{-1}^{1} x^{-2} d x=[-x-1]_{-1}^{1}=(-1)-1=-2

In Exercises 107–110, describe why the statement is incorrect. \(\int_{-2}^{1} \frac{2}{x^{3}} d x \geqslant[\frac{1}{x^{2}}]_{-2}^{1}=-\frac{3}{4}\) Text Transcription: \int_{-2}^{1} \frac{2}{x^{3}} d x \geqslant[\frac{1}{x^{2}}]_{-2}^{1}=-\frac{3}{4}

In Exercises 107–110, describe why the statement is incorrect. \(\int_{\pi / 4}^{3 \pi / 4} \sec ^{2} x d x=[\tan x]_{\pi / 4}^{3 \pi / 4}=-2\) Text Transcription: \int_{\pi / 4}^{3 \pi / 4} \sec ^{2} x d x=[\tan x]_{\pi / 4}^{3 \pi / 4}=-2

In Exercises 107–110, describe why the statement is incorrect. \(\int_{\pi / 2}^{3 \pi / 2} \csc x \cot x d x=[-\csc x]_{\pi / 2}^{3 \pi / 2}=2\) Text Transcription: \int_{\pi / 2}^{3 \pi / 2} \csc x \cot x d x=[-\csc x]_{\pi / 2}^{3 \pi / 2}=2

Buffon’s Needle Experiment A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is \(P=\frac{2}{\pi} \int_{0}^{\pi / 2} \sin \theta d \theta\) where \(\theta\) is the acute angle between the needle and any one of the parallel lines. Find this probability. Text Transcription: theta P=\frac{2}{\pi} \int_{0}^{\pi / 2} \sin \theta d \theta

Prove that \(\frac{d}{d x}\lfloor\int_{u(x)}^{v(x)} f(t) d t]=f(v(x)) v^{\prime}(x)-f(u(x)) u^{\prime}(x)\) Text Transcription: \frac{d}{d x}\lfloor\int_{u(x)}^{v(x)} f(t) d t]=f(v(x)) v^{\prime}(x)-f(u(x)) u^{\prime}(x)

True or False? In Exercises 113 and 114, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If F’(x) = G’(x) on the interval [a,b], then F(b)-f(a) = G(b) - G(a).

True or False? In Exercises 113 and 114, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f is continuous on [a,b], then f is integrable on [a,b]

Show that the function is constant for x>0 \(f(x)=\int_{0}^{1 / x} \frac{1}{t^{2}+1} d t+\int_{0}^{x} \frac{1}{t^{2}+1} d t\) Text Transcription: f(x)=\int_{0}^{1 / x} \frac{1}{t^{2}+1} d t+\int_{0}^{x} \frac{1}{t^{2}+1} d t

Find the function f(x) and all values of c such that \(\int_{c}^{x} f(t) d t=x^{2}+x-2\) Text Transcription: \int_{c}^{x} f(t) d t=x^{2}+x-2

Let \(G(x)=\int_{0}^{x}\left[s \int_{0}^{s} f(t) d t\right] d s\), where f is continuous for all real t. Find (a) G(0), (b) G’(0), (c) G’’(x), and (d) G’’(0) Text Transcription: G(x)=\int_{0}^{x}\left[s \int_{0}^{s} f(t) d t\right] d s