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MATH 113/114

PRACTICE FINAL

9:00 – 11:00

2 HOURS

No calculators

The last page is for scrap paper. NAME (first then last): ________________________________________________

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Multiple Choice: ID: ____________________ 1. Complete the following sentence with one of the given choices: Let f be a function with domain D. Then f has an absolute minimum value on D at a point c if:

(a) f(x) ≤ f(c) for all x in D.

(b) f(x) ≥ f(c) for all x in D.

(c) f(x) ≤ f(c) for all x in some open interval containing c.

(d) f(x) ≥ f(c) for all x in some open interval containing c. 2. The Fundamental Theorem of Calculus Part I implies that:

(a) The derivative of a function is the slope of the tangent line of the graph of the function.

(b) The integral of a function is the area between the graph of the function and the x-axis.

(c) Integration and differentiation are inverse processes.

(d) Every continuous function is integrable.

3. In the formula for Riemann sums, 1

lim ( )n

kn kA f x x

→∞ =

= ∆∑ , what does f(xk) represent?

(a) The height of the kth rectangle.

(b) The width of the kth rectangle.

(c) The area of the kth rectangle.

(d) The area under the curve between a and b. 4. The Fundamental Theorem of Calculus Part II says, “If F(x) is any antiderivative of f(x), then …”

(a) ( ) ( ) ( )b

af x dx F a F b= −∫ (b) ( ) ( ) ( )

b

aF x dx f a f b= −∫

(c) ( ) ( ) ( )b

af x dx F b F a= −∫ (d) ( ) ( ) ( )

b

aF x dx f b f a= −∫

3

5. Let 22( )

( 4)f x

xπ

=−

. Which of the following is TRUE?

(a) f(x) is increasing on its entire domain.

(b) f(x) is decreasing on its entire domain.

(c) f(x) is concave up on its entire domain.

(d) f(x) is concave down on its entire domain. Short Response:

1. Evaluate 3

22 1

x

d t dtdx

π

+∫ . __________________

2. What are the critical numbers of 4/5 2( ) ( 4)f x x x= − ? __________________

3. Evaluate 0

2

4

16 x dx−

−∫ . __________________

4. Evaluate 4

3

2( 1)

kk

=

+∑ . __________________

5. Find the function whose derivative is 2( ) secf x x′ =

That goes through the point ,74π⎛ ⎞⎜ ⎟⎝ ⎠

. __________________

Written Response: 1. (a) State the Extreme Value Theorem.

(b) Find the absolute minimum and absolute minimum of21( )

2xf xx

−=

+.

2. Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible.

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3. Sketch the graph of 2

2

2( )1

xf xx

=−

.

4. Use Riemann sums to calculate3

2

2( 2) x dx

−+∫ .

5. Integrate the following:

(a) 5

3 t t dtt−∫

(b) /2

2

04sin cos d

πθ θ θ⋅∫

(c) ( )102 33 7 9 x x dx+∫

(d) 2

3 2

01 x x dx+∫

6. Prove that the equation 2x – 1 – sin x = 0 has exactly one real root. 7. A plane flying horizontally at an altitude of 1 km and a speed of 500 km/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 km away from the station.