Math 105A/B Test One: Calculus Problems, Exams of Calculus

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Math 105A/B Test One

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1. Find the equation for the line that is tangent to the curve at the point (2, 80). f (x) = 10x + 15x^2
2. Let f (x) = x^2 + x. Use the limit definition to find f ′(x).

3. Let f (x) = |x|. a. Evaluate limx→ 0 f (x). b. Evaluate limx→ 0 f ′(x).
4. Let f (x) = 3 − (x − 1)^2 and I = [0, 2] a. Where on I does f (x) take its maximum values? b. Where on I does f (x) take its minimum values?

7. Explain why f ′′(x) = 6x − 2 implies that f has only one inflection point.
8. Let f (x) = x^2 + (^1) x + 2 for x > 0. a. Find the derivative of f (x). b. Find an antiderivative of f (x).

9.fencing as possible A gardener wants to enclose 1000 square meters of land. The garden is to be rectangular; one side, unfenced, lies using as little along a river. How much fencing is needed?

10.definition of derivative to prove Assume f (x) is differentiable and let g′(x) = f ′(x (^) ).g(x) = f (x) + a. Use the limit