Exam 3 Study Guide
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find an antiderivative of the given function. 1) x-3 + 1 4 x 2) cos πx + 6 sin x 6 Find the most general antiderivative. 6 3) ( t - t) dt
1)
2)
∫
3)
4)
∫ sin θ(cot θ + csc θ) dθ
4)
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 5) f(x) = x2 between x = 2 and x = 6 using the "midpoint rule" with four rectangles of equal 5) width. Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum ∑ f(ck) Δxk , using the indicated point in the kth k=1 subinterval for ck. 6) f(x) = x2 - 1, [0, 8], right-hand endpoint 6) y 56 52 48 44 40 36 32 28 24 20 16 12 8 4 -4 2 4 6 x
Find the formula and limit as requested. 7) For the function f(x) = 2x2+ 2, find a formula for the upper sum obtained by dividing the interval [0, 3] into n equal subintervals. Then take the limit as n→∞ to calculate the area under the curve over [0, 3].
7)
1
Evaluate the integral. 0 8) 3x2 + x + 3 dx
∫
8)
6 Find the derivative. x3 sin t dt 9) d dx 0
∫
9)
Find the total area of the region between the curve and the x-axis. 10) y = x2(x - 2)2; 0 ≤ x ≤ 2 Find the area of the shaded region. 11)
10)
11)
Evaluate the integral. 12)
∫ x2 ∫
x3 + 3 dx
12)
13)
sin t dt (8 + cos t)6
13)
Solve the problem. 14) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 32 cos 4t, v(0) = -10, s(0) = 12 Use the substitution formula to evaluate the integral. 4 9- x 15) dx x 1
14)
∫ ∫
15)
16)
π/2
cot x csc3 x dx
16)
π/6
2
Find the area of the shaded region. 17) y 25 20 15 10 5 -5 -4 -3 -2 -1 -5 -10 -15 -20 (-4, -24)-25 (0, 0)
1)
2)
∫
3)
4)
∫ sin θ(cot θ + csc θ) dθ
4)
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 5) f(x) = x2 between x = 2 and x = 6 using the "midpoint rule" with four rectangles of equal 5) width. Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum ∑ f(ck) Δxk , using the indicated point in the kth k=1 subinterval for ck. 6) f(x) = x2 - 1, [0, 8], right-hand endpoint 6) y 56 52 48 44 40 36 32 28 24 20 16 12 8 4 -4 2 4 6 x
Find the formula and limit as requested. 7) For the function f(x) = 2x2+ 2, find a formula for the upper sum obtained by dividing the interval [0, 3] into n equal subintervals. Then take the limit as n→∞ to calculate the area under the curve over [0, 3].
7)
1
Evaluate the integral. 0 8) 3x2 + x + 3 dx
∫
8)
6 Find the derivative. x3 sin t dt 9) d dx 0
∫
9)
Find the total area of the region between the curve and the x-axis. 10) y = x2(x - 2)2; 0 ≤ x ≤ 2 Find the area of the shaded region. 11)
10)
11)
Evaluate the integral. 12)
∫ x2 ∫
x3 + 3 dx
12)
13)
sin t dt (8 + cos t)6
13)
Solve the problem. 14) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 32 cos 4t, v(0) = -10, s(0) = 12 Use the substitution formula to evaluate the integral. 4 9- x 15) dx x 1
14)
∫ ∫
15)
16)
π/2
cot x csc3 x dx
16)
π/6
2
Find the area of the shaded region. 17) y 25 20 15 10 5 -5 -4 -3 -2 -1 -5 -10 -15 -20 (-4, -24)-25 (0, 0)