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18.01 ESG Independent Study Path

Unit 2 Practice Test

1. Use the definition of the limit to find

\frac{d}{dx}x^{2}

2. Find a tangent line to the curve y = 3x^{2} + 6x + 1 which passes through the point (0, 1).

3. Find y^{(7)} (= \frac{d^{7}y}{dx^{7}} ):

y = x^{7} + 3x^{6} + 4x^{5} − x^{3} + 78x^{2} − 5x + 4.

(Hint: think before starting; you don’t have to differentiate 7 times.)

4. Find \frac{dq}{dp}:

q = \frac{(p^{5} − p)(p^{3} + p^{2})}{p^{4} + (p^{3} + 1)(p^{3} − 1)}.

5. What is D_{x}[f(x)g(x)h(x)] equal to in terms of f′, g′, and h′ ? What is D_{x}[f(x)^{3}] ?

6. If f(2) = 3, f′(2) = −1, g(2) = −5, and g′(2) = 2, then what are (4f/g)′(2) and (f + fg − 1/g)′(2) ?