Practice questions

# Practice questions - at the point x y intersect the x-axis...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Fall 200 6

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18.01 Practice Questions for Exam 1 Solutions will be posted on the 18.01 website. No books, notes, or calculators will be allowed at the exam. 1. Evaluate each of the following, simplifying where possible; for (b) indicate reasoning. The letters a and k represent constants. d 3 t �� 3 u d 3 d 3 a ) b ) lim c ) sin kx d ) a + k sin 2 dt ln t u 0 tan 2 u dx 3 d� 2 e d 2. Derive the formula for x 3 at the point x = x 0 directly from the definition of dx derivative. 1 3 1 + h 3. Find lim by relating it to a derivative. (Indicate reasoning.) h 0 h 4. Sketch the curve y = sin 1 x, 1 x 1, and derive the formula for its derivative from that for the derivative of sin x . 5. For the function ax + b, x > 0 f ( x ) = , a and b constants , 1 x + x 2 , x 0 , a) find all values of a and b for which the function will be continuous; b) find all values of a and b for which the function will be differentiable. 6. For the curve given by the equation x 2 y + y 3 + x 2 = 8 , find all points on the curve where its tangent line is horizontal. 7. Where does the tangent line to the
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Unformatted text preview: ) at the point ( x , y ) intersect the x-axis? 8. The volume of a spherical balloon is decreasing at the instantaneous rate of − 10 cm 3 /sec, at the moment when its radius is 20 cm. At that moment, how rapidly is its radius decreasing? 9. Where are the following functions discontinuous? 1 + x 2 d a ) sec x b ) c ) x 1 − x 2 dx | | 10. A radioactive substance decays according to a law A = A e − rt , where A ( t ) is the amount in present at time t , and r is a positive constant. a) Derive an expression in terms of r for the time it takes for the amount to fall to one-quarter of the initial amount A . b) At the moment when the amount has fallen to 1/4 the initial amount, how rapidly is the amount falling? (Units: grams, seconds.)...
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