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TEST 1
Math 105 10/5/12 Name: (^) ︸ ︷︷ ︸ by writing my name I swear this work is my own
Read all of the following information before starting the exam:
- Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct).
- Circle or otherwise indicate your final answers.
- Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements. Put a smiley face next to your name for one point.
- This test has 5 problems and is worth 92 points, It is your responsibility to make sure that you have all of the pages!
- Good luck!
1. (16 points)
a. (4 pts) Draw the tangent line to the below graph at x = 1. Estimate the slope of the tangent line at x = 1.
0 1 2 3
1
The slope is equal to 1. (.6,.6) and (1.6,1.6), slope is 1.
b. (4 pts) Using an appropriate table of values, find lim x→ 1
2 x − 1 − 1 x − 1
x lim x→ 1
2 x − 1 − 1 x − 1 .9 1. .99 1. 1.001. 1.01.
lim x→ 1
2 x − 1 − 1 x − 1
c. (8 pts) Using the formal limit definition of the derivative ( lim h→ 0
, etc.), find f ′(1) for
f (x) =
2 x − 1
.
f ′(1) = lim h→ 0
f (1 + h) − f (1) h = lim h→ 0
2(1 + h) − 1 − 1 h = lim h→ 0
2(1 + h) − 1 − 1 h
2(1 + h) − 1 + 1 √ 2(1 + h) − 1 + 1
= lim h→ 0
(2(1 + h) − 1) − 1 h
= lim h→ 0
2 h h(
2(1 + h) − 1 + 1
= lim h→ 0
2(1 + h) − 1 + 1
4. (28 points) The following is a graph of f ′(x), NOT f (x). f (x) is a continuous function. Assume
the graph continues off to negative and positive infinity.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1
2
3 y=f'(x)
y=f''(x)
a. (4 pts) On which intervals is f (x) increasing or decreasing?
INC: (-4.4,-3) ∪ (-1.2,1.35) ∪ (2.8, ∞) DEC: (∞,-4.4) ∪ (-3,1.2) ∪ (1.35,2.8) b. (4 pts) On which intervals is f ′′(x) positive or negative?
POS: (−∞, 0) ∪ (2.2,∞) NEG: (0,2.2) c. (4 pts) Draw and label the graph of the f ′′(x) on the graph.
See above graph. d. (6 pts) At what x values does f (x) have local maximum or minimum? Identify which are maximums and which are minimums. (Hint: Clearly f (x) is not differentiable at x = −3. Since f (x) is continuous that means at x = −3, f (x) turns a sharp corner or comes to a point. We can’t define the derivative at such a point (think about the graph of |x|). Is x = −3 a local max, min, or neither? Consider it with the others.]
x=-4.4, -1.2, 2.8 are min, f ′′(x) > 0 for these points were f ′(x) = 0. x=1.35 is a max, f ′′(x) < 0 and f prime(x) = 0 at this x-value. x=-3 is a max, f ′(x) is positive before and negative after x=-3. e. (4 pts) At what x-values does f (x) have inflection points?
x=0,2.
f. (6 pts) Sketch and label 2 possible graphs of f (x) on the graph below. Remember, f (x) must be continuous.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1
2
3 y=f'(x) y=f(x)
y=f(x)
5. (8 points) The cubic f (x) =
x^3 3
− ax^2 + bx + c has an inflection point at x = 1. At x = 3 the
tangent line to the graph is y = 3x − 1. Determine a, b, and c.
f ′(x) = x^2 − 2 ax + b
f ′′(x) = 2x − 2 a
x = 1 is an inflection point implies that f ′′(1) = 0.
2(1) − 2 a = 0 → a = 1
At x = 3, the tangent line is y = 3x − 1. The slope of the tangent line is 3 when x=3. So, the derivative of the function at x = 3 is 3.
3 = (3^2 ) − 2(3) + b → 3 = 9 − 6 + b → b = 0
Finally, the tangent line touches the graph at the point (3,3*3-1) or (3,8). The point (3,8) is a point on the cubic.
− 32 + c → 8 = 9 − 9 + c → c = 8
The cubic is f (x) = x
3 3 −^ x
