Question: The velocity graph of a car accelerating
The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.
Transcribed Image Text:
םע (km/h) 80 40 20 30 (seconds) 10
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> The acceleration function (in m/s2 ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval. a(t) = t + 4, v(0) = 5, 0 ≤ t ≤ 10
> (a) Find an equation of the tangent line to the curve y = 2x sin x at the point (π/2, π). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
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> (a) If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b) If A(t) is the amount of the
> If f (x) = x2 - 4, 0 ≤ x ≤ 3, find the Riemann sum with n = 6, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.
> If f (x) = cos x 0 ≤ x ≤ 3 π/4 evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.
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> (a) If $1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b) Suppose $1000 is borrowed and
> Prove Property 3 of integrals. Property 3 of integrals: cf(x) dx = c f(x) dx, where c is any constant
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> A freshly brewed cup of coffee has temperature 958C in a 20°C room. When its temperature is 70°C, it is cooling at a rate of 1°C per minute. When does this occur?
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> In a murder investigation, the temperature of the corpse was 32.5°C at 1:30 pm and 30.3°C an hour later. Normal body temperature is 37.0°C and the temperature of the surroundings was 20.0°C. When did the murder take place?
> (a) Find the Riemann sum for f (x) = 1/x, 1 ≤ x ≤ 2, with four terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a)
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> A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The
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> Differentiate. y = sinθ cosθ
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> Dinosaur fossils are too old to be reliably dated using carbon-14. (See Exercise 11.) Suppose we had a 68-millionyear- old dinosaur fossil. What fraction of the living dinosaur’s 14C would be remaining today? Suppose the minimum detectable amount is 0.1%
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> The table shows speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida. (a) Estimate the distance the race car traveled during this time period using the velocities at the be
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> With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. (On a TI use the Is. command or a For-EndFor loop, on a Casio use Isz
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> Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation ab
> Find dy/dx by implicit differentiation. cos(xy) = 1 + sin y
> Copy the vectors in the figure and use them to draw the following vectors. (a). u + v (b). u + w (c) v + w (d). u - v (e). v + u + w (f). - w - v V
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> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 2, 3, 0 , b = 1, 0,5
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> Identify and sketch the graph of each surface. 4x2 + 4y2 - 8y + z2 = 0
> Identify and sketch the graph of each surface. y2 + z2 = 1 + x2
> Identify and sketch the graph of each surface. -4x2 + y2 - 4z2 = 4
> Identify and sketch the graph of each surface. 4x - y + 2z = 4
> Find a vector equation and parametric equations for the line. The line through the point (2, 2.4, 3.5) and parallel to the vector 3i + 2j - k
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> The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after t years. (b) How much of the sample remains after 100 years? (c) After how long will only 1 mg remain?
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> Find parametric equations for the line. The line through (1, 0, -1) and parallel to the line 1 3 (x – 4) = 1 2 y = z + 2
> Find parametric equations for the line. The line through (4, -1, 2) and (1, 1, 5)
> Find the magnitude of the torque about P if a 50-N force is applied as shown. 50 N 30° 40 cm P
> A constant force F = 3i + 5j + 10k moves an object along the line segment from (1, 0, 2) to (5, 3, 8). Find the work done if the distance is measured in meters and the force in newtons.
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> Describe and sketch the surface. z = sin y
> Describe and sketch the surface. xy = 1
