Find the length of the curve. x = t3, y = 32 t2, 0 ≤ t ≤ 3 The length of the curve x = t3, y = 32 t2 on 0 ≤ t ≤ 3 is (Type an integer or a fraction. )
Find the length of the following curve. x = (2t+7)3/23, y = 3t+t2 2, 0 ≤ t ≤ 2 The length of the curve is . (Simplify your answer. )
Find the area of the surface generated by revolving the curve x = 12 cos(2t), y = 5+12 sin(2t) on 0 ≤ t ≤ π2 about the x-axis. The area of the surface generated by revolving the curve x = 12 cos(2t), y = 5+12 sin(2t) on 0 ≤ t ≤ π2 is square units. (Type an exact answer in terms of π. )
Compare the relative rates at which 17x+2 and 4 x+1 grow as x→∞ by comparing their growth rates with x as x→∞. What can be said about the relative rates at which the functions f(x) = 17x+2 and g(x) = 4x+1 grow? Choose the correct answer below. A. (x) = 17x+2 and g(x) = 4x+1 grow at the same rate as x→∞. B. f(x) = 17x+2 grows slower than g(x) = 4x+1 as x→∞. C. f(x) = 17x+2 grows faster than g(x) = 4x+1 as x→∞.
Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d2 ydx2 at this point. x = 1 t+5, y = tt−5, t = 6 Write the equation of the tangent line. y = x − (Simplify your answers. Use integers or fractions for any numbers in the expression. ) What is the value of d2y dx2 at this point? d2y dx2 = (Type an integer or a simplified fraction. )
Determine if the following functions grow faster, slower, or at the same rate as the function e6 x as x→∞. a. x−3 b. 2 x2+s i n2 x c. 3 x d. 66 x e. (52)6 x f. ex6 g. e6 x6 h. log5x a. The function x−3 grows slower than e6 x as x→∞ because limx→∞e6 xx−3 = b. The function 2 x2+s i n2 x grows slower than e6 x as x→∞ because limx→∞e6 x2 x2+s i n2 x = c. The function 3 x grows slower than e6 x as x→∞ because limx→∞e6 x3 x = e. The function (52)6 x grows slower than e6 x as x→∞ because limx→∞e6 x(52)6 x = f. The function ex6 grows slower than e6 x as x→∞ because limx→∞e6 xex6 = g. The function e6 x6 grows at the same rate as e6 x as x→∞ because limx→∞e6 xe6 x6 = h. The function log5x grows slower than e6 x as x→∞ because limx→∞e6 xlog5x =
Determine if the following functions grow faster, slower, or at the same rate as the function x2 as x→∞. a. x2+3 x b. x5−x2 c. x4+x3 d. (x+4)2 e. 4 xlnx f. 4 x g. 4 x3 e−x h. 8 x2 a. The function x2+3 x grows at the same rate as x2 as x→∞ because limx→∞x2 x2+3 x = b. The function x5−x2 grows faster than x2 as x→∞ because limx→∞x2 x5−x2 = c. The function x4+x3 grows at the same rate as x2 as x→∞ because limx→∞x2 x4+x3 = d. The function (x+4)2 grows at the same rate as x2 as x→∞ because limx→∞x2(x+4)2 = e. The function 4 xlnx grows slower than x2 as x→∞ because limx→∞x24 xlnx = . f. The function 4 x grows faster than x2 as x→∞ because limx→∞x24 x = . g. The function 4 x3 e−x grows slower than x2 as x→∞ because limx→∞x24 x3 e−x = . h. The function 8 x2 grows at the same rate as x2 as x→∞ because limx→∞x28 x2 =
Determine if the following functions grow faster, slower, or at the same rate as the function lnx as x→∞ a. log7x b. ln5 x c. lnx d. 3 x e. 3 x f. 6 lnx g. 4 x h. ex a. The function log7x grows at the same rate as lnx as x→∞ because limx→∞lnxlog7x = b. The function ln5 x grows at the same rate as lnx as x→∞ because limx→∞lnxln5 x = c. The function lnx grows at the same rate as lnx as x→∞ because limx→∞lnxlnx = d. The function 3 x grows faster than lnx as x→∞ because limx→∞lnx3 x = e. The function 3 x grows faster than lnx as x→∞ because limx→∞lnx3 x = f. The function 6 l nx grows at the same rate as l nx as x→∞ because limx→∞lnx6 l nx = g. The function 4 x grows slower than lnx as x→∞ because limx→∞lnx4 x = h. The function ex grows faster than lnx as x→∞ because limx→∞lnxex =
Determine if the following functions grow faster, slower, or at the same rate as the function l nx as x→∞. a. log7(x2) b. log1212 x c. 13 x d. 6 x2 e. x−5 lnx f. e−x g. ln(lnx) h. ln(4 x+7) a. The function log7(x2) grows at the same rate as lnx as x→∞ because limx→∞lnxlog7(x2) = b. The function log1212 x grows at the same rate as lnx as x→∞ because limx→∞lnxlog1212 x = c. The function 13 x grows slower than lnx as x→∞ because limx→∞lnx13 x = d. The function 6 x2 grows slower than lnx as x→∞ because limx→∞lnx6 x2 = e. The function x−5 lnx grows faster than lnx as x→∞ because limx→∞lnxx−5 lnx = . f. The function e−x grows slower than lnx as x→∞ because limx→∞lnxe−x =. g. The function ln(lnx) grows slower than lnx as x→∞ because limx→∞lnxln(lnx) = h. The function ln(4 x+7) grows at the same rate as lnx as x→∞ because limx→∞lnxln(4 x+7) =
The equation below gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = 3 t−3, y = 9 t2; −∞ < t < ∞ Find a Cartesian equation for the particle's path. y = Graph the Cartesian equation below. Indicate the direction of motion as t increases. Choose the correct graph below. A. B. C. Indicate the portion of the graph traced by the particle. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. None of the graph B. The entire graph C. The portion of the graph where t is less than D. The portion of the graph where t is greater than
The parametric equations and parameter intervals for the motion of a particle in the xy-plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = sint and y = 5 cos(2t), −π2 ≤ t ≤ π2 Find an equation that relates x and y directly. y = Graph the parametric curve below. Indicate the direction of motion as t increases. Choose the correct graph below. A. B. C. Indicate the portion of the graph traced by the particle. Choose the correct answer below. A. The portion of the graph where t is greater than B. None of the graph C. The entire graph D. The portion of the graph where t is less than
Given parametric equations and parameter intervals for the motion of a particle in the xy-plane below, identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = −3 sec(t), y = 3 tan(t), −π2 < t < π2 Choose the correct answer for the Cartesian equation representing the same path defined by the given parametric equations. A. (x−y)2 = 18 B. (x−y)2 = 9 C. x2−y2 = 18 D. x29−y29 = 1 Choose the correct graph that represents this motion. A. B.
The curve with parametric equations x = t, y = 1−cost, 0 ≤ t ≤ 2π is called a sinusoid and is shown in the accompanying figure. Find the point (x, y) where the slope of the tangent line is (a) largest and (b) smallest. (a) Begin by finding dydx. dydx = Thus the slope of the tangent line can be found by the function s i nt. Determine where s i nt is largest on the interval 0 ≤ t ≤ 2π. The slope of the tangent line, s i nt, is largest at t = (Type an exact answer, using π as needed. ) Determine the point (x, y) where the slope is the largest. The point where the slope is largest is (Type an ordered pair. Type an exact answer, using π as needed. ) (b) Recall that dydx = s i nt. Determine where s i nt is smallest on the interval 0 ≤ t ≤ 2π. The slope of the tangent line, s i nt, is smallest at t = (Type an exact answer, using π as needed. ) Determine the point (x, y) where the slope is smallest. The point where the slope is the smallest is (Type an ordered pair. Type an exact answer, using π as needed. )
Consider the following statement. There are real numbers a and b such that a+b = a+b. To prove the statement, it suffices to find values of a and b that satisfy the property. Show that you can do this by entering appropriate values for a and b in the box below, as an ordered, comma-separated list of two numbers.
Consider the following statement. There is an integer n such that 2n2 − 5n + 2 is prime. To prove the statement it suffices to find a value of n such that (n, 2n2 − 5n + 2) satisfies the property " 2n2 − 5n + 2 is prime. " Show that you can do this by entering appropriate values for n and 2 n2 − 5n + 2
Disprove the following statement by giving a counterexample. (Enter your answers as a comma-separated list. ) For every integer p, if p is prime then p2 − 1 is even.