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Math 105: Review for Exam I
- Let f(x) = 3 +
x + 5. (a) What is the natural domain of f? (b) What is the range of f?
- For the graph of f shown, answer the following.
(a) Evaluate the following. i. f′(−2) ii. f(3) iii. (^) xlim→ 3 − f(x) iv. (^) xlim→ 3 + f(x) v. (^) xlim→ 3 f(x) vi. (^) xlim→ 2 f(x) (b) Where is f discontinuous? (c) Where does f′^ fail to exist?
2
4
−
−4 −2 2 4
f(x)
3
3
- Let f(x) = 3x^2 − 2 x.
(a) Compute the average rate of change of f on the interval [2, 2 .1].
(b) Using the limit definition of the derivative, find f′(x).
(c) Find the equation of the tangent line to f at x = 2.
(d) How would the derivative of g(x) = f(x) + 5 compare to f′(x)?
(e) How would the derivative of h(x) = 5f(x) compare to f′^ (x)?
- Fill in the table showing the graphical relationships between f, f′, and f′′. f positive negative increasing decreasing concave up concave down f′ f′′
- Given the graph of f, sketch a graph of f′^.
f(x)
f ‘(x)
- Shown below is a graph of f′^ on its entire domain. The graph is NOT f.
At which x-value(s) (a) does f have a stationary point? (b) does f have a local max? (c) does f have a local min? (d) does f′^ have a stationary point? (e) does f′^ have a local max? (f) does f′^ have a local min? (g) is f greatest? (h) is f least? (i) is f′^ greatest? (j) is f′^ least? (k) is f′′^ greatest? (l) is f′′^ least? On what interval(s) is (a) f increasing?
(b) f decreasing? (c) f′^ increasing? (d) f′^ decreasing? (e) f concave up? (f) f concave down?
f ’(x)
a b c
d
e f g h i jk
- You are constructing a box that will have a square bottom (with sides of length x) and no top. Furthermore, the sum of x and the height of the box must be 30 inches. (a) Write a formula for the volume of the box as a function of x.
(b) What is the domain of your function?
(c) What value of x maximizes the volume of the box? What is this maximum volume?
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