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mathematics
calculus

Questions and Answers of Calculus

Prove that if (x f(x)) is a point of inflection of the graph of f and f" exists in an open interval that contains c, then f"(c) = 0.
Show that the function g(x) = x|x| has an inflection point at (0, 0) but g"(0) does not exist.
Suppose f is differentiable on an interval I and f'(x) > 0 for all numbers in except for a single number c. Prove that f is increasing on the entire interval I.
The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f,g, and h whose values at 0 are all 0 and, for
(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. F(x) = 2x3 + 3x2 - 36x
Given thatwhich of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.
Use the graph of f and g and their tangent lines at (2, 0) to find f(x)/g(x).
Use a graph to estimate the value of the limit. Then use 1'Hospital's Rule to find the exact value.
Illustrate 1'Hospital's Rule by graphing both f(x)/g(x) and f'(x)/g'(x) near x = 0 to see that these ratios have the same limit as x → 0. Also, calculate the exact value of the limit.F(x) = ex - 1,
Find the limit. Use 1'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If 1'hospital's Rule doesn't apply, explain why.(a)(b) (c) (d)
What happens if you try to use 1'Hospital's Rule to find the limit? Evaluate the limit using another method.
Investigate the family of curves f(x) = ex - cx. In particular, find the limits as x → ± ∞ and determine the values of for which has an absolute minimum. What happens to the minimum points as
If an initial amount A0 of money is invested at an interest rate r compounded times a year, the value of the investment after t years isIf we let n †’ ˆž, we refer to the continuous compounding
(a) By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 8. In each case sketch
Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if x*1 is a left or right endpoint. (For instance, in Maple use
The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she
Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table. Find lower and upper estimates
The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.
Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. a. f(x) = 2x/(x2+1), 1 ≤ x ≤ 3 b. f(x) = √sinx, 0 ≤ x ≤ π
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
Let A be the area under the graph of an increasing continuous function f from a to b, and let Ln and Rn be the approximations to A with n subintervals using left and right endpoints, respectively.(a)
(a) Express the area under the curve y = x5 from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).
Find the exact area under the cosine curve y = cosx from x = 0 to x = b, where 0 ≤ b ≤ π/2. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is
(a) Estimate the area under the graph of f(x) = cos x from x = 0 to x = π/2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an
(a) Estimate the area under the graph of f(x) = 1 + x2 from x = -1 to x = 2 using three rectangles and right end points. Then improve your estimate by using six rectangles. Sketch the curve and the
Evaluate the upper and lower sums for f(x) = 2 + six x, 0 ‰¤ x ‰¤ Ï€, with n = 2, 4 and 8. Illustrate with diagrams like Figure 14.
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
Evaluate the Riemann sum for f(x) = 3 - 1/2x, 2 ≤ x ≤ 14, with six subintervals, taking the sample points to be left endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.
If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles (use RiemannSum or middlesum and middlebox commands in Maple), check the answer to Exercise 11 and
Use a calculator or computer to make a table of values of right Riemann sums Rn for the integralWith n = 5, 10, 50, and 100. What value do these numbers appear to be approaching?
Express the limit as a definite integral on the given interval.a.b.
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.a.b.
Express the integral as a limit of Riemann sums. Do not evaluate the limit.
If f(x) = ex - 2, 0 ≤ x ≤ 2, find the Riemann sum with n = 4 correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a
Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit.
The graph of is shown. Evaluate each integral by interpretingit in terms of areas.a.b. c. d.
Evaluate the integral by interpreting it in terms of areas.a.b.c.
Use the result of Example 3 to evaluate
Write as a single integral in the form
IfAnd Find
The graph of a function f is given. EstimateUsing five subintervals with (a) Right endpoints, (b) Left endpoints, (c) Midpoints.
For the function whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning.A.B.C.D.E. f'(1)
Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of
Use the properties of integrals to verify the inequality without evaluating the integrals.a.b.
Use Property 8 to estimate the value of the integral.a.b. c.
Let f(x) = 0 if is any rational number and f(x) = 1 if is any irrational number. Show that f is not integrable on [0, 1]
A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for
Express the limit as a definite integral.Consider f(x) = x4
Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal placesa.b.
Explain exactly what is meant by the statement that "differentiation and integration are inverse processes."
Evaluate the integral.a.b. c.
LetWhere f is the function whose graph is shown (a) Evaluate g(0), g(1), g(2), g(3), and g(6). (b) On what interval is g increasing? (c) Where does g have a maximum value? (d) Sketch a rough graph of
What is wrong with the equation?a.b.
Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. a. y = 3√x, 0 ≤ x ≤ 27 b. y = sin x, 0 ≤ x ≤ π
Sketch the area represented by g(x). Then find g'(x) in two ways:(a) By using Part 1 of the Fundamental Theorem(b) By evaluating the integral using Part 2 and then differentiating.
Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.
Find the derivative of the function.a.b.
On what interval is the curveconcave downward?
The Fresnel function S was defined in Example 3 and graphed in Figures 7 and 8.(a) At what values of does this function have local maximum values?(b) On what intervals is the function concave
LetWhere f is the function whose graph is shown (a) At what values of x do the local maximum and minimum values of g occur? (b) Where does g attain its absolute maximum value? (c) On what intervals
Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.a.b. c.
Justify 3 for the case h < 0.
(a) Show that for 1 ‰¤ ˆš(1+x3) ‰¤ 1 + x3 for x ‰¥ 0.(b) Show that
Show thatBy comparing the integrand to a simpler function
Find a function f and a number such thatFor all x > 0
A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate f = f(t) , where is the time measured in months since its last overhaul. Because a fixed cost A is
Verify by differentiation that the formula is correct. a. ∫ 1/(x2√(1+x2)) dx = - (√1+x2)/x + C b. ∫ cox3x dx = sinx -1/3 sin3x + C
Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. ∫ (cox x + 1/2 x) dx
Evaluate the integral.a.b. c.
Use a graph to estimate the -intercepts of the curve y = 1 - 2x -5x4. Then use this information to estimate the area of the region that lies under the curve and above the x-axis.
Find the general indefinite integral. a. ∫ (x2 + x-2) dx b. ∫ (x4 - 1/2 x3 + 1/4x - 2) dx c. ∫ (u + 4)(2u + 1) du
If Ï‰'(t) is the rate of growth of a child in pounds per year, what doesrepresent?
If oil leaks from a tank at a rate of r(t) gallons per minute at time t, what doesrepresent?
In Section 4.7 we defined the marginal revenue function R'(x) as the derivative of the revenue function R(x), where x is the number of units sold. What doesrepresent?
If is measured in meters and f(x) is measured in Newton, what are the units for
The velocity function (in meters per second) is given for a particle moving along a line. Find (a) The displacement (b) The distance traveled by the particle during the given time interval. v(t) =
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. (b) The distance traveled during the given time
The linear density of a rod of length 4 m is given by ((x) = 9 + 2√x measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.
The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car.
The marginal cost of manufacturing x yards of a certain fabric is C'(x) = 3 - 0.01x + 0.000006x2 (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to
A bacteria population is 4000 at time t = 0 and its rate of growth is 1000 . 2' bacteria per hour after t hours. What is the population after one hour?
Shown is the power consumption in the province of Ontario, Canada, for December 9, 2004 (P is measured in megawatts; t is measured in hours starting at midnight). Using the fact that power is the
Evaluate the integral by making the given substitution. a. ∫ e-x dx, u = -x b. ∫ x2 √(x3 + 1) dx, u = x3 +1 c. ∫ cos3θ sin θ dθ, u = cos θ
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). a. ∫ x(x2 -1)3 dx b. ∫ ecosx sin x dx
Evaluate the definite integral.a.b. c.
Evaluate the indefinite integral. a. ∫ x sin(x2) dx b. ∫ (1-2x)9 dx c. ∫ (x+1) √(2x + x2) dx
Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. y = √(2x + 1) , 0 ≤ x ≤ 1
Evaluateby writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.
Which of the following areas are equal? Why?
An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t) = 100e-0.01t liters per minute. How much oil leaks out during the first hour?
The volume of inhaled air in the lungs at time t is
If f is continuous on R, prove that
If a and b are positive numbers, show that
Use Exercise 90 to evaluate the integral
Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be(a) Left endpoints(b) Midpoints. In each case draw a diagram and explain what the Riemann sum
Evaluateby interpreting it in terms of areas.
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0)
Use a graph to give a rough estimate of the area of the region that lies under the curve y = x √x, 0 ≤ x ≤ 4. Then find the exact area.
Find the derivative of the function.a.b.

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