Math I Assignment Not Graded I Final Review, FRQ’s
Name:
Date:
Graded Assignment
Final Review, FRQ’s
(9 points)
Free Response Question 1 (with calculator)
You may use your calculator for this question
1. The region R is bounded by the x-axis, y-axis, x = 3 and y = 1/√(x+1).
A. Find the area of region R.
B. Find the volume of the solid formed when the region R is revolved about the x-axis
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Math I Assignment Not Graded I Final Review, FRQ’s
C. The solid formed in part B is divided into two solids of equal volume by a plane perpendicular to the
x-axis. Find the x-value where this plane intersects the x-axis.
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Math I Assignment Not Graded I Final Review, FRQ’s
(9 points)
Free Response Question 2 (with calculator)
You may use a calculator on this question.
2. In this problem, you will investigate the family of functions h(x) = x + cos(ax) , where a is a positive constant
such that 0 < a < 4.
A. Graph the curves y(x) = x + cos(¼x) and y(x) = x + cos(4x) in the space provided below:
B. For what values of a will h(x) have a relative maximum at x = 1?
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Math I Assignment Not Graded I Final Review, FRQ’s
C. For what value(s) of a will h(x) have an inflection point at x = 1?
D. Which values of a make h(x) strictly decreasing? Justify your answer.
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Math I Assignment Not Graded I Final Review, FRQ’s
(9 points)
Free Response Question 3 (with calculator)
You may use your calculator for this question
3. Water usage rate for Seattle in a given 24-hour period is represented in the table and graph below
(Note: t = 0 corresponds to midnight). Time is measure in hours, and the rate of water usage is
measured in millions of gallons per hour.
A. Using three trapezoids of equal width, estimate
Explain the physical meaning of this definite integral.
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Math I Assignment Not Graded I Final Review, FRQ’s
B. Does the estimate in part A overestimated or underestimate the value of
on the interval 0 < t < 8? Explain your reasoning.
C. Using your answer from part A, estimate the average water usage over the 24-hour period.
D. Estimate the slope of the usage rate curve at t = 12. Describe the method that you use.
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Math I Assignment Not Graded I Final Review, FRQ’s
(9 points)
Section 4 Free Response (no calculator)
Free-Response Question 4 (no calculator)
You may not use your calculator for this question.
4. The function f (x) has the value f (−1) = 1. The slope of the curve y = f (x) at any point is given by the
expression dy/dx = (2x + 1)( y + 1)
A. Write an equation for the line tangent to the curve y = f (x) at x = − 1.
B. Use the tangent line from part A to estimate f (−0.9)
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Math I Assignment Not Graded I Final Review, FRQ’s
C. Use separation of variables to find an explicit or implicit formula for y = f (x), with no integrals remaining.
D. Find lim f (x)
x ∞
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Math I Assignment Not Graded I Final Review, FRQ’s
(9 points)
Free Response Question 5 (no calculator)
You may not use your calculator for this question.
5. The figure at right is the graph of f ′′ (x),
the second derivative of a function f (x).
The domain of the function f (x) is all real
numbers, and the graph shows
f ′′ (x) for −2.6≤ x ≤3.6.
A. Find all values of x in the interval (−2.6, 3.6) where f ′(x) has a horizontal tangent.
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Math I Assignment Not Graded I Final Review, FRQ’s
B. Find all values of x in the interval (−2.6, 3.6) where f (x) is concave upwards. Explain your answer.
C. Suppose it is known that in the interval (−3.6, 3.6), f (x) has critical points at x =1.37, and x = −0 .97.
Classify these points as relative maxima or minima of f (x). Explain your answer.
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Math I Assignment Not Graded I Final Review, FRQ’s
(9 points)
Free Response Question 6 (no calculator)
You may not use your calculator for this question.
6. Water is flowing at the rate of 50 m 3/min into a holding tank shaped like a cone, sitting vertex down. The
tank’s base diameter is 40 m and a height of 10 m.
A. Write an expression for the rate of change of the water level with respect to time, in terms of h (the
water’s height in the tank).
B. Assume that, at t = 0, the tank of water is empty. Find the water level, h, as a function of the time t.
C. What is the rate of change of the radius of the cone with respect to time when the water is 8 feet deep?
Your Score
___ of 54
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