Name: Class: Date:

Name: ________________________ Class: ___________________ Date: __________

ID: A

M1B T1 Sp 09 1. [5 points each] Decide if the statement is true or false (not always true). For the true statements provide a proof or state an appropriate theorem and for the false statements explain clearly or give a counter example (an example for which the statement is not true). Your counter examples can be graphs or algebraic examples.

È ˘ a) The area bounded by the graph of a function f and the x-axis on a closed interval ÍÍÎ a,b ˙˙˚ is

È ˘ b) If F is a continuous function over the interval ÍÍÎ a,b ˙˙˚ and is defined by F(x) = ÍÈÍ a , b ˙˘˙ , then F ′(x) = f(x). Î ˚

∫

∫

b

a

d)

∫

π

b

f(c)dx = ∫ f(x)dx a

sec 2 x dx = tan(π ) − tan(0)

0

1

f(x)dx .

a

x

f(t)dt for every x in

a

È È ˘ ˘ c) For a continuous function f on a closed interval ÍÍÎ a , b ˙˙˚ , there exists a c in ÍÍÎ a , b ˙˙˚ such that:

∫

b

Name: ________________________

ID: A

È ˘ e) For every continuous functions f and g over the interval ÍÍÎ a , b ˙˙˚ , if È ˘ for all x in ÍÍÎ a , b ˙˙˚ .

∫

b

a

f(x)dx =

∫

b

g(x)dx then f(x) = g(x)

a

. 2. [8 points] For the function F(x) =

∫

x

2

ln(t 3 + 4)dt find an equation of the tangent line at x = 2.

4

. 3. [7 points] Use the velocity function v(t) = 30e −t / 4 and the initial position s(0) = −1 to calculate the final position at t = 4.

.

2

Name: ________________________

ID: A

4. [7 points each] Evaluate the indicated integral.

∫

2 − x dx =

a) x

b)

∫

3x + 20

c)

∫

x

d)

∫

x+7

dx =

1 − x4

π/2

π/4

dx =

cot(x)dx =

.

3

Name: ________________________

ID: A

5. [7 points] Find a value of c that satisfies the conclusion of the Integral Mean Value Theorem for the function f(x) = x 2 − 1 over the interval [1, 3].

. 6. a) [7 points] Set up the Riemann sum that approximates the area under the graph of y = tan−1 (x) over the interval [1, 3] using left-endpoint evaluation with a regular partition of size 16. (set up and simplify but do not calculate completely)

b) [2 points] Does this sum overestimate or underestimate the actual area under the curve?

c) [3 points] True/False: f ave <

π 4

4

Name: ________________________

ID: A

7. a) [8 points] Use Simpson’s rule to approximate the value of

∫

π/2

−π / 2

cos(x) dx with n = 4.

b) [5 points] Set up and simplify, but do not calculate, the Trapezoidal Rule to approximate the above integral with the same n value.

.

5