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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Evaluate the definite integral. Use a graphing utility to verify your result. (π/2 J-π/2 (2t + cos t) dt
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Find the area of the given region.y = x - x2 −14 y 1 X
Find the area of the given region.y = cos x 1 y 4 IN 2
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Find the area of the given region.y = 1/x2 1 y I 2 X
Find the area of the given region.y = x + sin x 4 3 2 y la 2 X
Find the particular solution of the differential equation that satisfies the initial condition(s).g′(x) = 4x2, g(−1) = 3
A function f is defined below. Use geometric formulas to find ∫80 f (x) dx. f(x)= = (4, x < 4 lx, x ≥ 4
(a) Use a graphing utility to graph a slope field for the differential equation,(b) Use integration and the given point to find the particular solution of the differential equation, and(c) Graph the
Find the particular solution of the differential equation that satisfies the initial condition(s). h′(x) = 7x6 + 5, h(1) = −1
The graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. To print an enlarged copy of the graph, go to MathGraphs.com. -2 -1 2 1 y -2 -²
Use the graph of f′ shown in the figure to answer the following. (a) Approximate the slope of f at x = 4. Explain.(b) Is f(5) − f(4) > 0? Explain.(c) Approximate the value of x where f is
Find the area of the region bounded by the graphs of the equations.y = 5x2 + 2, x = 0, x = 2, y = 0
A function f is defined below. Use geometric formulas to find ∫120 f (x) dx. f(x) : = 6, x6 -x + 9, x 6
Find the particular solution of the differential equation that satisfies the initial condition(s). f ″(x) = 3x2, f′(−1) = −2, f(2) = 3
Find the area of the region bounded by the graphs of the equations.y = x3 + 6x, x = 2, y = 0
Find the area of the region bounded by the graphs of the equations.y = 1 + 3√x, x = 0, x = 8, y = 0
Find the area of the region bounded by the graphs of the equations.y = 2√x - x, y = 0
The graphs of f and f′ each pass through the origin. Use the graph of f ″ shown in the figure to sketch the graphs of f and f′. To print an enlarged copy of the graph, go to MathGraphs.com.
Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.y = 25 − x2, [1, 4]
Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.y = 3x − 2, [2, 5]
Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.y = 5x2 + 1, [0, 2]
Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.y = 27 - x3, [1, 3]
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 - x2, [-2, 2]
The functionis defined on [0, 1], as shown in the figure. Show thatdoes not exist. Does this contradict Theorem 4.4? Why or why not? THEOREM 4.4 0, x = 0 f(x) = 1 I >x>0
Use a graph to explain why ∫aa f (x) dx = 0, if f is defined at x = a.
Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.y = x2 - x3, [-1, 1]
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. "b "b [ * [f(x) + g(x)] dx = = [°² f(x) dx + [². g(x) dx
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. • dx = [ f* f(x) dx ] [ [ * 8(x) dx] [*f(x)g(x) dx =
Describe two ways to evaluate ∫31 (x + 2) dx.Verify that each method gives the same result.
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) = x4 + 7, [0, 2]
Use a graph to explain why ∫ba kf (x) dx = k ∫ba f (x) dx, if f is integrable on [a, b] and k is a constant.
Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.y = 2x3 - x2, [1, 2]
Use the limit process to find the area of the region bounded by the graph of the function and the y-axis over the given y-interval. Sketch the region.y = 3y - y2, 2 ≤ y ≤ 3
Consider an n-sided regular polygon inscribed in a circle of radius r. Join the vertices of the polygon to the center of the circle, forming n congruent triangles (see figure).(a) Determine the
Find the constants a and b, where a || [ ₁ (x − 4) dx = So |x - 4| dx = 20. 16 and
The area A between the graph of the functionand the t-axis over the interval [1, x] is(a) Find the horizontal asymptote of the graph of g.(b) Integrate to find A as a function of x. Does the graph of
Consider a particle moving along the x-axis, where x(t) is the position of the particle at time t, x′(t) is its velocity, and x″(t) is its acceleration.x(t) = t3 − 6t2 + 9t − 2, 0 ≤ t ≤
Find F as a function of x and evaluate it at x = 0, x = π/4, and x = π/2.F(x) = ∫x-π sinθ dθ
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
Show that the functionis constant for x > 0. (1/x x 1 1 = S² R² + ₁ ² + ² + ₁ ² dt 1 dt 1 0 f(x) =
Use the Second Fundamental Theorem of Calculus to find F'(x). ) - f₁ √t csc t dt F(x) =
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The value of ∫ba f (x) dx must be positive.
What is the value of n?(a)(b) 11 Σ i=1 5(5 + 1) 2
A function is continuous, nonnegative, concave upward, and decreasing on the interval [0, a]. Does using the right endpoints of the subintervals produce an overestimate or an underestimate of the
Explain why the Midpoint Rule almost always results in a better area approximation in comparison to the endpoint method.
Does the Midpoint Rule ever give the exact area between a function and the x-axis? Explain.
What are the index of summation, the upper bound of summation, and the lower bound of summation for 8 Σ ( – 4)? i=3
(a) Integrate to find F as a function of x,(b) Demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).F(x) = ∫x4 t3/2 dt
Find the function f (x) and all values of c such that f* f(t) dt = x² + x - 2.
Letwhere f is continuous for all real t. Find(a) G(0),(b) G'(0),(c) G''(x), (d) G''(0). G(x) = S[$ f* 0 f(1) f(t) dt ds
Evaluate the definite integral. Use a graphing utility to verify your result. S²₁ -1 (7-3t) dt
A teacher places n seats to form the back row of a classroom layout. Each successive row contains two fewer seats than the preceding row. Find a formula for the number of seats used in the layout.
Evaluate, if possible, the integral ∫20 [[x]] dx.
Evaluate the definite integral. Use a graphing utility to verify your result. (6x² – 3x) dx
Why isconsidered an accumulation function? - 5.50 F(x) = f(t) dt
Find the general solution of f′(x) = −2x sin x2.
Evaluate the definite integral. Use a graphing utility to verify your result. L (2t - 1)² dt
Evaluate the definite integral. Use a graphing utility to verify your result. 3 [² (1/2-1) dx
Evaluate the definite integral. Use a graphing utility to verify your result. L₁ -1 (1²5) dt
Write a definite integral that represents the area of the region. f(x) = x2 TT 4 3 نیا 2 y 2 نیا X
Write a definite integral that represents the area of the region. f (x) = 25 - x2 15 10 5 4 -6 -4 -2 У + 246
Evaluate the definite integral. Use a graphing utility to verify your result. JI (8x³ - x) dx
What does it mean for a function F to be an antiderivative of a function f on an interval I?
Write a definite integral that represents the area of the region. f (x) = 4/ x2 + 2 - 1 1 y 1 X x=
Evaluate the definite integral. Use a graphing utility to verify your result. [(u - 12/2) du -2
Evaluate the definite integral. Use a graphing utility to verify your result. 8 -8 x¹/3 dx
Explain how to find the area of a region using a definite integral in your own words.
Evaluate the definite integral. Use a graphing utility to verify your result. Sus Ju u-2 du
Write a definite integral that represents the area of the region. (Do not evaluate the integral.)f (x) = cos x y 4 2 X
Evaluate the definite integral. Use a graphing utility to verify your result. 8. La X ax
Write a definite integral that represents the area of the region. (Do not evaluate the integral.)g(y) = y3 4 3 2 1 y 2 4 6 8 00
Evaluate the definite integral. Use a graphing utility to verify your result. x - √√x 3 S = dx
Evaluate the definite integral. Use a graphing utility to verify your result. 1 -1 (√1 - 2) dt
Evaluate the definite integral. Use a graphing utility to verify your result. 5²66- (6-1) √t dt
Find the indefinite integral and check the result by differentiation.∫(9x8 − 2x − 6) dx
Evaluate the definite integral. Use a graphing utility to verify your result. Co J-1 G (11/3 - 12/3) dt
Evaluate the definite integral. Use a graphing utility to verify your result. |2x - 5| dx
Evaluate the definite integral. Use a graphing utility to verify your result. -1 x-x² -8 23√x dx
Evaluate the definite integral. Use a graphing utility to verify your result. xp |6 - zx| 4 Jo -
Evaluate the definite integral. Use a graphing utility to verify your result. 10 |x² - 4x + 31 dx
Evaluate the definite integral. Use a graphing utility to verify your result. [₁² (3 (3 - x - 3) dx
Find the indefinite integral and check the result by differentiation.∫1/x5 dx
Evaluate the definite integral. Use a graphing utility to verify your result. So Jo (sin x-7) dx
Evaluate the definite integral. Use a graphing utility to verify your result. * /4 10 1 - sin20 0 -502 ᎾᏢ -
Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0).∫40 x dx
Find the indefinite integral and check the result by differentiation.∫(2 − 3/x10) dx
Evaluate the definite integral. Use a graphing utility to verify your result. St (2 + cos x) dx
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0).∫4-3 9 dx
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Evaluate the definite integral. Use a graphing utility to verify your result. n/4 sec²0 tan²0 + 1 -de
Evaluate the definite integral. Use a graphing utility to verify your result. π/2 S Jπ/4 (2 - csc²x) dx
Evaluate the definite integral. Use a graphing utility to verify your result. 9/1J -π/6 sec² x dx
Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Evaluate the definite integral. Use a graphing utility to verify your result. n/3 -π/3 4 sec 0 tane de
Find the indefinite integral and check the result by differentiation.∫(sin x − 6 cos x) dx
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