Cost engineers at NASA have the task of projecting the cost P of major space projects

In Exercises 14, refer to the function f whose graph is shown in Figure 1. Estimate L4 and M4 on [0, 4].

In Exercises 14, refer to the function f whose graph is shown in Figure 1. Estimate R4, L4, and M4 on [1, 3].

In Exercises 14, refer to the function f whose graph is shown in Figure 1. Find an interval [a, b] on which R4 is larger than  b a f (x) dx. Do the same for L4.

In Exercises 14, refer to the function f whose graph is shown in Figure 1. Justify 3 2  2 1 f (x) dx 9 4 .

In Exercises 58, let f (x) = x2 + 3x. Calculate R6, M6, and L6 for f on the interval [2, 5]. Sketch the graph of f and the corresponding rectangles for each approximation.

In Exercises 58, let f (x) = x2 + 3x. Use FTC I to evaluate A(x) =  x 2 f (t) dt

In Exercises 58, let f (x) = x2 + 3x. Find a formula for RN for f on [2, 5] and compute  5 2 f (x) dx by taking the limit.

In Exercises 58, let f (x) = x2 + 3x. Find a formula for LN for f on [0, 2] and compute  2 0 f (x) dx by taking the limit.

Calculate R5, M5, and L5 for f (x) = (x2 + 1)1 on the interval [0, 1].

Let RN be the Nth right-endpoint approximation for f (x) = x3 on [0, 4] (Figure 2). (a) Prove that RN = 64(N + 1)2 N2 . (b) Prove that the area of the region within the right-endpoint rectangles above the graph is equal to 64(2N + 1) N2 y 32 64 x 1234 FIGURE 2 Approximation RN for f (x) = x3 on [0, 4]

Which approximation to the area is represented by the shaded rectangles in Figure 3? Compute R5 and L5. x y 30 18 6 12345

Calculate any two Riemann sums for f (x) = x2 on the interval [2, 5], but choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.

In Exercises 1316, express the limit as an integral (or multiple of an integral) and evaluate. 13. lim N 6N N j=1 sin  3 + j 6N

In Exercises 1316, express the limit as an integral (or multiple of an integral) and evaluate.lim N 3 N N 1 k=0  10 + 3k N

In Exercises 1316, express the limit as an integral (or multiple of an integral) and evaluate.lim N 5 N N j=1  4 + 5j/N

In Exercises 1316, express the limit as an integral (or multiple of an integral) and evaluate.lim N 1k + 2k ++ Nk Nk+1 (k > 0)

In Exercises 1730, calculate the indefinite integral. 17.   4x3 2x2 dx

In Exercises 1730, calculate the indefinite integral.x9/4 dx

In Exercises 1730, calculate the indefinite integral.]sin( 8)d

In Exercises 1730, calculate the indefinite integral.cos(5 7)d

In Exercises 1730, calculate the indefinite integral.(4t 3 12t 4)dt

In Exercises 1730, calculate the indefinite integral.(9t 2/3 + 4t 7/3)dt

In Exercises 1730, calculate the indefinite integral.sec2 x dx

In Exercises 1730, calculate the indefinite integral.tan 3 sec 3 d

In Exercises 1730, calculate the indefinite integral.(y + 2) 4 dy

In Exercises 1730, calculate the indefinite integral.3x3 9 x2 dx

In Exercises 1730, calculate the indefinite integral.(ex x) dx

In Exercises 1730, calculate the indefinite integral.e4x dx

In Exercises 1730, calculate the indefinite integral.4x1 dx

In Exercises 1730, calculate the indefinite integral.sin(4x 9)dx

In Exercises 3136, solve the differential equation with the given initial condition. 31. dy dx = 4x3, y(1) = 4

In Exercises 3136, solve the differential equation with the given initial condition. dy dt = 3t 2 + cost, y(0) = 12

In Exercises 3136, solve the differential equation with the given initial condition. dy dx = x1/2, y(1) = 1

In Exercises 3136, solve the differential equation with the given initial condition. dy dx = sec2 x, y  4  = 2

In Exercises 3136, solve the differential equation with the given initial condition. dy dx = ex , y(0) = 3

In Exercises 3136, solve the differential equation with the given initial condition. dy dx = e4x , y(1) = 1

Find f (t) if f (t) = 1 2t, f (0) = 2, and f (0) = 1.

At time t = 0, a driver begins decelerating at a constant rate of 10 m/s2 and comes to a halt after traveling 500 m. Find the velocity at t = 0

In Exercises 3942, use the given substitution to evaluate the integral. 39.  2 0 dt 4t + 12 , u = 4t + 12

In Exercises 3942, use the given substitution to evaluate the integral.(x2 + 1)dx (x3 + 3x)4 , u = x3 + 3x

In Exercises 3942, use the given substitution to evaluate the integral./6 0 sin x cos4 x dx, u = cos x

In Exercises 3942, use the given substitution to evaluate the integral.sec2(2 )tan(2)d, u = tan(2 )

In Exercises 4392, evaluate the integral. 43.  (20x4 9x3 2x) dx

In Exercises 4392, evaluate the integral.2 0 (12x3 3x2)dx

In Exercises 4392, evaluate the integral.(2x2 3x)2 dx

In Exercises 4392, evaluate the integral.1 0 (x7/3 2x1/4)dx

In Exercises 4392, evaluate the integral.x5 + 3x4 x2 dx

In Exercises 4392, evaluate the integral.3 1 r4 dr

In Exercises 4392, evaluate the integral.3 3 |x2 4| dx

In Exercises 4392, evaluate the integral.4 2 |(x 1)(x 3)| dx

In Exercises 4392, evaluate the integral.3 1 [t] dt

In Exercises 4392, evaluate the integral.2 0 (t [t]) 2 dt

In Exercises 4392, evaluate the integral.(10t 7) 14 dt

In Exercises 4392, evaluate the integral.3 2  7y 5 dy

In Exercises 4392, evaluate the integral.(2x3 + 3x) dx (3x4 + 9x2)5

In Exercises 4392, evaluate the integral.1 3 x dx (x2 + 5)2

In Exercises 4392, evaluate the integral..  5 0 15x x + 4 dx

In Exercises 4392, evaluate the integral.t 2 t + 8 dt

In Exercises 4392, evaluate the integral.1 0 cos  3 (t + 2)  dt

In Exercises 4392, evaluate the integral. /2 sin 5 6 d

In Exercises 4392, evaluate the integral.t 2 sec2(9t 3 + 1)dt

In Exercises 4392, evaluate the integral.sin2(3 ) cos(3)d

In Exercises 4392, evaluate the integral..  csc2(9 2)d

In Exercises 4392, evaluate the integral.sin 4 cos d

In Exercises 4392, evaluate the integral./3 0 sin cos2/3 d

In Exercises 4392, evaluate the integral.sec2 t dt (tan t 1)2

In Exercises 4392, evaluate the integral.e92x dx

In Exercises 4392, evaluate the integral.3 1 e4x3 dx

In Exercises 4392, evaluate the integral.x2ex3 dx

In Exercises 4392, evaluate the integral.ln 3 0 exex dx

In Exercises 4392, evaluate the integral.ex 10x dx

In Exercises 4392, evaluate the integral.e2x sin(e2x )dx

In Exercises 4392, evaluate the integral.ex dx (ex + 2)3

In Exercises 4392, evaluate the integral.sin cos ecos2 +1 d

In Exercises 4392, evaluate the integral./6 0 tan 2 d

In Exercises 4392, evaluate the integral.2/3 /3 cot 1 2 d

In Exercises 4392, evaluate the integral.dt t (1 + (ln t)2)

In Exercises 4392, evaluate the integral.cos(ln x) dx x

In Exercises 4392, evaluate the integral.e 1 ln x dx x

In Exercises 4392, evaluate the integral.dx x ln x

In Exercises 4392, evaluate the integral.dx 4x2 + 9

In Exercises 4392, evaluate the integral.0.8 0 dx  1 x2

In Exercises 4392, evaluate the integral.12 4 dx x  x2 1

In Exercises 4392, evaluate the integral.3 0 x dx x2 + 9

In Exercises 4392, evaluate the integral.3 0 dx x2 + 9

In Exercises 4392, evaluate the integral.dx  e2x 1

In Exercises 4392, evaluate the integral.x dx  1 x4

In Exercises 4392, evaluate the integral.5/ 2 5/ 2 dx  25 x2

In Exercises 4392, evaluate the integral.4 0 dx 2x2 + 1

In Exercises 4392, evaluate the integral.8 5 dx x  x2 16

In Exercises 4392, evaluate the integral.1 0 (tan1 x)3 dx 1 + x2

In Exercises 4392, evaluate the integral.cos1 t dt  1 t2

Combine to write as a single integral:  8 0 f (x) dx +  0 2 f (x) dx +  6 8 f (x) dx

Let A(x) =  x 0 f (x) dx, where f is the function shown in Figure 4. Identify the location of the local minima, the local maxima, and points of inflection of A on the interval [0, E], as well as the intervals where A is increasing, decreasing, concave up, or concave down. Where does the absolute maximum of A occur? x y A B C DE

Find the local minima, the local maxima, and the inflection points of A(x) =  x 3 t dt t2 + 1

Aparticle starts at the origin at time t = 0 and moves with velocity v(t) as shown in Figure 5. (a) How many times does the particle return to the origin in the first 12 seconds? (b) What is the particles maximum distance from the origin? (c) What is the particles maximum distance to the left of the origin? 2 4 4 2 5 10 v(t) (m/s) t (s)

On a typical day, a city consumes water at the rate of r(t) = 100 + 72t 3t2 (in thousands of gallons per hour), where t is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 pm and midnight?

The learning curve in a certain bicycle factory isL(x) = 12x1/5 (in hours per bicycle), which means that it takes a bike mechanic L(n) hours to assemble the nth bicycle. If a mechanic has produced 24 bicycles, how long does it take her or him to produce a subsequent batch of 12?

Cost engineers at NASA have the task of projecting the cost P of major space projects. It has been found that the cost C of developing a projection increases with P at the rate dC/dP 21P 0.65, where C is in thousands of dollars and P in millions of dollars. What is the cost of developing a projection for a project whose cost turns out to be P = $35 million?

An astronomer estimates that in a certain constellation, the number of stars per magnitude m, per degree-squared of sky, is equal to A(m) = 2.4 106m7.4 (fainter stars have higher magnitudes). Determine the total number of stars of magnitude between 6 and 15 in a 1-degree-squared region of sky.

Evaluate  8 8 x15 dx 3 + cos2 x , using the properties of odd functions.

Evaluate  1 0 f (x) dx, assuming that f is an even continuous function such that  2 1 f (x) dx = 5,  1 2 f (x) dx = 8

Plot the graph of f (x) = sin mx sin nx on [0, ] for the pairs (m, n) = (2, 4), (3, 5) and in each case guess the value of I =  0 f (x) dx. Experiment with a few more values (including two cases with m = n) and formulate a conjecture for when I is zero.

Show that  x f (x) dx = xF (x) G(x) where F (x) = f (x) and G (x) = F (x). Use this to evaluate  x cos x dx.

Prove 2  2 1 2x dx 4 and 1 9  2 1 3x dx 1 3

Plot the graph of f (x) = x2 sin x, and show that 0.2  2 1 f (x) dx 0.9.

Find upper and lower bounds for  1 0 f (x) dx, for y = f (x) in Figure 6.

In Exercises 108113, find the derivative. 108. A (x), where A(x) =  x 3 sin(t3)dt

In Exercises 108113, find the derivativeA ( ), where A(x) =  x 2 cost 1 + t d

In Exercises 108113, find the derivatived dy  y 2 3x dx

In Exercises 108113, find the derivativeG (x), where G(x) =  sin x 2 t 3 dt

In Exercises 108113, find the derivativeG (2), where G(x) =  x3 0 t + 1 dt

In Exercises 108113, find the derivativeH (1), where H (x) =  9 4x2 1 t dt

Explain with a graph: If f is increasing and concave up on [a, b], then LN is more accurate than RN . Which is more accurate if f is increasing and concave down?

Explain with a graph: If f is linear on [a, b], then the  b a f (x) dx = 1 2 (RN + LN ) for all N.

In this exercise, we prove x x2 2 ln(1 + x) x (for x > 0) 1 (a) Show that ln(1 + x) =  x 0 dt 1 + t for x > 0. (b) Verify that 1 t 1 1 + t 1 for all t > 0. (c) Use (b) to prove Eq. (1). (d) Verify Eq. (1) for x = 0.5, 0.1, and 0.01.

Let F (x) = x  x2 1 2  x 1  t2 1 dt Prove that F (x) and y = cosh1 x differ by a constant by showing that they have the same derivative. Then prove they are equal by evaluating both at x = 1.

Let f be a positive increasing continuous function on [a, b], where 0 a

How can we interpret the quantity I in Eq. (2) if a

The isotope thorium-234 has a half-life of 24.5 days. (a) What is the differential equation satisfied by y(t), the amount of thorium-234 in a sample at time t? (b) At t = 0, a sample contains 2 kg of thorium-234. How much remains after 40 days?

The Oldest Snack Food? In Bat Cave, New Mexico, archaeologists found ancient human remains, including cobs of popping corn whose C14-to-C12 ratio was approximately 48% of that found in living matter. Estimate the age of the corn cobs.

The C14-to-C12 ratio of a sample is proportional to the disintegration rate (number of beta particles emitted per minute) that is measured directly with a Geiger counter. The disintegration rate of carbon in a living organism is 15.3 beta particles per minute per gram. Find the age of a sample that emits 9.5 beta particles per minute per gram.

What is the interest rate if the PV of $50,000 to be delivered in 3 years is $43,000?

An equipment upgrade costing $1 million will save a company $320,000 per year for 4 years. Is this a good investment if the interest rate is r = 5%? What is the largest interest rate that would make the investment worthwhile?Assume that the savings are received as a lump sum at the end of each year.

Find the PV of an income stream paying out continuously at a rate of 5000e0.1t dollars per year for 5 years, assuming an interest rate of r = 4%.

Calculate the limit: (a) lim n  1 + 4 n n (b) lim n  1 + 1 n 4n (c) lim n  1 + 4 n 3n