Calculus, Fall 2010

Midterm Exam 1

Mar 30 2010, 10:10AM

1. Find lim_{y→+∞}y sin \frac{2}{y}.

2. (a): Write down the definition of “f(x) is continuous at x = 0” in terms of ϵ and δ.

(b): Use the ϵ and δ argument to show that if lim_{x→1}f(x) = 3 and lim_{x→1}g(x) = 4, then

lim_{x→1}(f(x) + g(x)) = 7.

3. State (need not prove) the Intermediate Value Theorem. Use it to prove that y = x^{3} + x^{2} + 3x + 1 has at least a real root.

4. Find an approximate value of \frac{1}{\sqrt{2}} sin(0.24π) using linear approximation. Show that the error of the approximation is smaller than 5 × 10^{−4}.

Hint: To get a good linear approximation, it is important to select the center point of approximation, x_{0}.

5. Use implicit differentiation to find y′ and y′′ at (1, 1) where y(x) is implicitly given by x^{4} + y^{2} = 2.

6. Show how Newton’s method can be used to find \sqrt[3]{3}. (State the procedure but need not give the numerical value).

7. Practice on Find dy/dx for the following functions. Need not simplify your expression.

(a): y = \sqrt{x + \sqrt{x}}, x > 0.

(b): y = sin ( cos^{2}(x))

8. Graph f(x) = x^{1/3}(x − 4). Indicate all critical points and points of reflection.