MIT OpenCourseWare: 18.01 Single Variable Calculus - Practice Questions for Exam 1, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

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 e^2 tan^2 u^ dx^3 d�

d

2. Derive the formula for x 3 at the point x = x 0 directly from the definition of dx derivative.

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 graph of y = f (x) at the point (x 0 , y 0 ) 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 0 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 0.

b) At the moment when the amount has fallen to 1/4 the initial amount, how rapidly

is the amount falling? (Units: grams, seconds.)