Mathematics 36, section 5 – AP Calculus Information on Final Exam December 6, 1999

Background

The final exam for this section of AP Analysis will be given at 8:30 a.m. on Saturday, December 11, in our regular classroom – Beaven 118 . The exam will be comprehensive, covering all the material we have studied since the beginning of the semester. I will write the exam to be roughly twice the length of the in-class midterms. However, you will have the full three-hour period 8:30 - 11:30 a.m. to work on the exam if you need that much time. The questions will be similar to those on the 3 inclass exams. I may include some questions asking for definitions of terms or statements of theorems.

Topics to be Covered

1) Functions defined by graphs, tables, formulas, the “library of functions”

2) Derivatives (know the definition): know how to sketch the graph of the derivative of a function defined by a graph, how to approximate values of the derivative given a table for the function, how to use the derivative rules on functions defined by formulas.

3) The meaning of the signs of f′(x), f′′(x).

4) Differentiability = local linearity, tangent lines, etc.

5) Taylor polynomials, approximations, the error bound

6) Taylor series,

7) Total change of a function, Riemann sums, the definite integral (know the definition), antiderivatives

8) The Fundamental Theorem of Calculus, and

9) Integration by substitution, by parts, using the table. A copy of the table of integrals from the text will be provided with the exam; any integral you need to compute will be do-able by some combination of substitution, integration by parts, and/or consultation with the table).

10) Applications of integration (setting up problems via Riemann sums; in limit a definite integral is obtained) – volumes by slices, arclengths, physical examples like work, mass from non-constant density functions, etc.

11) Differential equations – slope fields and solutions

12) Solving differential equations via separation of variables and integration, including population growth models.

13) Parametric curves.

Suggestion on How to Study

Begin by reviewing the class notes and the discussion write-ups and lab reports from your groups’ work. Look at the in-class midterm problems and try to work out solutions for those again, without refering to your previous work. Look over some of the problems from the text from the previous review sheets. Then take a couple of hours and try the practice exam problems below.

I. A) (10) One of the following tables gives a function that is approximately exponential:

given by ca^{x}. Identify which one it is and give values for c and a:

\begin{matrix}x & 0 & .2 & .4 & .6 & .8 \\ g(x) & 3.10 & 3.40 & 3.70 & 4.00 & 4.30 \end{matrix}  \begin{matrix}x & 0 & .2 & .4 & .6 & .8 \\ h(x) & 3.10 & 3.56 & 4.09 & 4.70 & 5.40 \end{matrix}

B) (10) The following graph shows y = f(x) for some function. Sketch the graph y = 2f(x − 2) + 1.

II. The following graph shows y = f(x). A) (10) Sketch the graph y = f′(x) qualitatively. Show any points where f′(x) does not

exist by hollow circle(s) at those x’s. B) (10) Let g(x) be the antiderivative of f(x) with g(0) = 0. Sketch the graph y = g(x)

qualitatively.

III. A) (5) What is the precise definition of the derivative of a function f at x = a? B) (5) What is the precise definition of the definite integral of a function f over an interval

[a, b]? C) (10) Give a precise statement of the Fundamental Theorem of Calculus.

IV. All parts of this question refer to the function f(x) = sin(x). A) (10) Compute the Taylor polynomial p_{5}(x) of f(x) at a = 0, and use it to approximate

f(.6). B) (10) Using the Error Bound for Taylor polynomial approximation, determine the small-

est degree polynomial you would need to approximate f(.6) to 7-decimal place accuracy. (You do not need to compute the approximation.)

V. A circular metal plate of uniform thickness has total radius 6 inches. The density of the metal decreases with radius according to the following table:

r in inches  0  1  2  3  4 5 6
density in grams/square inch 200 190 170 140 100 90 80

A) (15) Approximate the total mass of the plate using a suitable Riemann sum. B) (5) Do you think your estimate is larger or smaller than the true mass? Explain,

including any assumptions you are making.

VI. Compute the following integrals. A) (10) ∫_{0}^{1} cos(πx) − x^{2}e^{x^{3}} dx B) (10) ∫ x^{3} sin(3x) dx C) (10) ∫ \frac{x−2}{\sqrt{4−x^{2}}} dx

VII. Both parts of this question refer to the region R in the first quadrant bounded by the graph y = 4 − x^{2}, the x-axis, and the y-axis. A) (10) Find the volume of the solid obtained by revolving R around the x-axis. B) (10) Same, but revolving around the line x = 3.

VIII. In a chemical reaction, the amount y = y(t) of a certain compound that is present changes over time and satisfies the differential equation

\frac{dy}{dt} = k(y − 2)(y − 6)

for some k > 0. A) (10) Taking k = 1, sketch the slope field for this equation, showing the slopes at points

with y = 1, 2, 3, 4, 5, 6, 7. B) (5) Suppose y(0) = 5. What happens to the amount of the compound as t → ∞? C) (15) Solve the differential equation to obtain a formula for y(t). (Hint: you will need

to use an entry from the table of integrals for this. State the one you use for partial credit.)

IX. Consider the parametric curve (x(t), y(t)) = (cos(2t), sin(t)). A) (10) Sketch the curve for t ∈ [−π, π]. B) (5) Derive an xy-equation for this curve by eliminating t. (Hint: cos(A + B) =

cos(A) cos(B) − sin(A) sin(B).) C) (5) Is every point (x, y) that satisfies the equation from B on the curve? Why or why

not?