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Math 106: Exam I Prep - Area, Integration, Approximations, and Diff. Equations, Exams of Calculus

Birla Institute of Technology and ScienceCalculus

Review questions for exam i of math 106, covering topics such as finding areas under curves, decreasing and concave functions, approximations using various methods, integrals, and differential equations. Students are asked to find the area under sinx on [0, π], integrate e^(√x) dx, x^3 dx, and cos(3(5x)) dx. They must also order quantities related to a decreasing and concave up function and find approximations for the integral of f(x) = 12/(1 + x^2) dx using left, right, midpoint, trapezoidal, and simpson's rules. Additionally, students are asked to find bounds for errors in approximating the integral of ln(x) dx, estimate y(2.5) using euler's method, find the arc length of y = 1 - x^2, and write integrals for the volume of a rotated region and the work done in pumping fluid in a pyramid.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

parul
parul 🇮🇳

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Math 106: Review for Exam I
1. Find the following.
(a) the area under y= sin xon the interval [0,π]
(b) Ze√x
√xdx
(c) Zx3
1+x8dx
(d) Zcos3(5x)dx
2. If f(x) is decreasing and concave up, put the following quantities in ascending order.
L100,R100,T100,M100,Zb
a
f(x)dx
What can you say with certainty about where S200 would fit into your list above?
3. Find the best possible left, right, midpoint, trapezoidal, and Simpson’s approximations to Z12
4
f(x)dx
given the data in the table below.
x4 6 8 10 12
f(x) 15 11 8 4 3
pf3

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Math 106: Review for Exam I

  1. Find the following. (a) the area under y = sin x on the interval [0, π]

(b)

∫ (^) e√x √x dx

(c)

∫ (^) x 3 1 + x^8 dx

(d)

cos^3 (5x) dx

  1. If f(x) is decreasing and concave up, put the following quantities in ascending order.

L 100 , R 100 , T 100 , M 100 ,

∫ (^) b a^ f(x)^ dx What can you say with certainty about where S 200 would fit into your list above?

  1. Find the best possible left, right, midpoint, trapezoidal, and Simpson’s approximations to

given the data in the table below.^4 f(x)^ dx x 4 6 8 10 12 f(x) 15 11 8 4 3

  1. Find bounds for each of the following errors if I =

2 ln^ x dx. (a) |I − L 100 |

(b) |I − T 100 |

(c) |I − M 100 |

(d) |I − S 100 |

  1. Use Euler’s method with three steps on the differential equation dy dx = y − x to estimate y(2.5) if y(1) = 0.
  2. Find the arc length of y = √ 1 − x^2 on the interval [0, 1].