Calculus I Practice Exam 2A

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1 Calculus I Practice Exam 2A This practice exam emphasizes conceptual connections and understanding to a greater degree than the exams that are usually administered in introductory single-variable calculus courses. It is designed to guide students who are taking such courses to a deeper mastery of the material. While a number of questions here are fairly typical for actual examinations, you should not infer from the expression practice exam that exams encountered in introductory singlevariable calculus courses will ask the same types of questions. Multiple choice Use the table on the right to answer questions and 2.. If H(x) = f(x), what is H ()? g(x) x f(x) g(x) f (x) g (x) a. 5 4 b. 7 9 c. 7 9 d If H = g f, what is H ()? a. 20 b. 23 c. 35 d Find the tangent slope of the curve x 3 y + y 2 = at (,). a. dy dx = 0 b. dy dx = 3 c. dy dx = d. dy dx = 206 R. Boerner, School of Mathematical & Statistical Sciences Arizona State University

2 4. An ice cube has a side of centimeter. You put it into the sun, and it starts melting. It loses volume at a constant rate of 0.4 cubic centimeter every minute but maintains a cubic shape. At what rate is the side shrinking after minute, in centimeters per minute? Round your answer to 3 decimal places. a b c d Before the New Horizons probe reached the dwarf planet Pluto in July 205, the radius of Pluto was not known with high accuracy. The previous best value was 84 kilometers, with an uncertainty of 20 kilometers. If we make the simplifying assumption that Pluto is a perfect sphere, what is the uncertainty in its volume that corresponds to the uncertainty in its radius? Use linear approximation and round appropriately. You will not get credit for the exact answer. a km 3 b km 3 c. 346 million km 3 d. 352 million km 3 e. 358 million km 3 6. Evaluate the it: a. b. 0 c. 0 0 d. x6 x5 x x 5 x 4 7. The differentiable, invertible function f(x) has the tangent y = 2x + 3 at x =. Find the tangent of f (x) at x = 5. a. y = 2x+3 b. y = 2 x + 3 c. y = 2 x 3 2 d. y = 2 x R. Boerner, School of Mathematical & Statistical Sciences Arizona State University

3 8. Find the 99 th derivative of y = sin (5x). a. y (99) = 5 99 cos (5x) b. y (99) = 5 99 cos (5x) c. y (99) = 5 99 sin (5x) d. y (99) = 5 99 sin (5x) e. None of the above Calculus I Practice Exam 2A 9. Use linear approximation to approximate 226 and round to 6 digits. a. 5 b c d Find and simplify the derivative of f(x) = tan ( x 2). a. 2 tan 2 ( x 2) x 3 b. 2x x 4 + c. 2 x 3 +x 2 d. x 3 +x Free response. Find the tangent line to the curve f(x) = e x at x = 0 and use your result to approximate e 0.0. Compare to the calculator value. 2. Use L Hospital s Rule to evaluate the it exactly: ( x + x ln x ) 3. Use L Hospital s Rule to evaluate the it exactly: x x x. 206 R. Boerner, School of Mathematical & Statistical Sciences Arizona State University

4 Answers: Multiple Choice: B 2C 3D 4A 5D 6A 7C 8B 9B 0B Free Response:. The equation of the tangent line of a function f at x = a is L(x) = f(a) + f (a)(x a) In this case, a = 0 and f (x) = e x, so f (a) =. Therefore, the tangent is L(x) = + x. Since the tangent approximates the function value near the point of attachment, e 0.0 L(0.0) =.0. This is very close to the calculator value of Since the given it is an indeterminate form of type, we have to turn the difference into a suitable quotient before we can apply L Hospital s rule. We do this by creating a common denominator and combining the two fractions. ( x + x ln x ) = ln x (x ) x + ln x (x ) The new it is an indeterminate form of type 0, so L Hospital s rule applies: 0 ln x (x ) x + ln x (x ) = x + x ln x + (x ) x This it is again an indeterminate form of type 0 0, so we use L Hospital s rule again. (We facilitate taking the denominator derivative by distributing the /x in the second term). After that, we can find the it by evaluation. 206 R. Boerner, School of Mathematical & Statistical Sciences Arizona State University

5 x ln x + = x x (x ) x + ln x + = x 2 x + x x + = 2 x 2 x + 3. Since the given it is an indeterminate form of type 0, we have to use the technique we learned in class to be able to apply L Hospital s rule: We now evaluate the it of the exponent: x x = e x ln x x x = ln x x x ln x x This it is an indeterminate form of type, so we can use L Hospital s rule: ln x x x = x x = x x = 0. It follows that the original it we were asked to evaluate is e 0 =. Note: while this guide is being made freely available to ASU students and the general public for personal use, it is not to be uploaded to third-party websites, especially not ones that profit from such content. If you found this document on a third-party website such as Course Hero or Chegg, the document is being served to you in violation of copyright law. 206 R. Boerner, School of Mathematical & Statistical Sciences Arizona State University