18.01 Practice Exam 1

1. (5 points each) Find the derivative of each of the following functions:

(a) f(x) = x^{7} + 7x;

(b) f(x) = x/(x^{4} + 1);

(c) f(x) = 3x^{3}(x^{3} + 3)^{3};

(d) f(x) = x sin x;

(e) f(x) = e^{x^{3}};

(f) f(x) = cos(e^{x} + x^{1/3});

(g) f(x) = \sqrt{x^{4} + 4};

(h) f(x) = x + ln(x^{2} + 1);

(i) f(x) = 2^{x};

(j) f(x) = cos(xe^{\sqrt{x}}).

2. (5 points each) Evaluate the following limits (or explain why the limit does not exist):

(a) lim_{x→1} \frac{x^{2} − 1}{(x − 1)^{2}} ;

(b) lim_{x→0} \frac{(x − 1) sin x}{x}.

3. (10 points) Use the definition of the derivative to find f′(1) if f(x) = 1/x.

4. (10 points) An object moving on a straight line has position s = t^{3} − t^{2}.

(a) Find the velocity v and acceleration a.

(b) Which direction (positive, negative or neither) is the object moving when t = 1?

5. (10 points) Consider the function f(x) = ^{3} \sqrt{x}.

(a) Is f(x) continuous at x = 0?

(b) Is f(x) differentiable at x = 0?

If so, then find the value of the derivative.

If not, then state whether the graph has a tangent line at the point (0, 0).

6. (10 points) Find the slope of the tangent line to the graph of y^{2} +2xy = e^{x} at the point (0, 1).