MTH 263 Test 2 Study Guide Fall 2020
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use implicit differentiation to find dy/dx. 1) x3 + 3x2y + y3 = 8 1)
Provide an appropriate response. 2) Use the Intermediate Value Theorem to prove that 3x3 + 9x2 - 3x + 5 = 0 has a solution
between -4 and -3. 2)
Use implicit differentiation to find dy/dx and d2y/dx2. 3) 4 y - y = 2x 3)
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested.
4) 6x2y - cos y = 7 , tangent at (1, ) 4)
1
Provide an appropriate response. 5) Use a calculator to graph the function f to see whether it appears to have a continuous
extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be?
f(x) = 6x - 1
x
5)
6) At the two points where the curve x2 - xy + y2 = 4 crosses the x-axis, the tangents to the curve are parallel. What is the common slope of these tangents?
6)
7) Explain why the following five statements ask for the same information. (a) Find the roots of f(x) = 4x3 - 3x - 4. (b) Find the x-coordinate of the points where the curve y = 4x3 crosses the line y = 3x + 4. (c) Find all the values of x for which 4x3 - 3x = 4. (d) Find the x-coordinates of the points where the cubic curve y = 4x3 - 3x crosses the line y = 4. (e) Solve the equation 4x3 - 3x - 4 = 0.
7)
8) Find the normal to the curve x2 + y2 = 2x + 2y that is parallel to the line y + x = 0. 8)
2
9) Find a value of c that will make
f(x) =
sin2 4x x2
, x 0
c, x = 0 continuous at x = 0.
9)
10) Graph y = -tan x and its derivative together on - 2
, 2
. Is the slope of the graph of
y = -tan x ever positive? Explain.
10)
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 11) s = 8t2 + 4t + 7, 0 t 2
Find the body's displacement and average velocity for the given time interval. 11)
3
Find the derivative of the function.
12) f(t) = (2 - t)(2 + t3)-1 12)
Find an equation of the tangent line at the indicated point on the graph of the function. 13) w = g(z) = 4 + 6 - z, (z, w) = (5, 5) 13)
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 14) s = - t3 + 8t2 - 8t, 0 t 8
Find the body's speed and acceleration at the end of the time interval. 14)
Divide numerator and denominator by the highest power of x in the denominator to find the limit.
15) lim x
16x2
3 + 9x2 15)
16) lim x
3 x + x-1 4x - 2
16)
4
17) lim t
9t2 - 27 t - 3
17)
18) lim x
5x + 3
3x2 + 1 18)
Find the limit.
19) lim x 0-
1 x1/5
+ 8 19)
20) lim x
( 7x2 + 3 - 7x2 - 3) 20)
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Answer Key Testname: MTH 263 TEST 2 STUDY GUIDE FALL 2020.TMP
1) - x2 + 2xy x2 + y2
2) Let f(x) = 3x3 + 9x2 - 3x + 5 and let y0 = 0. f(-4) = -31 and f(-3) = 14. Since f is continuous on [-4, -3] and since y0 = 0 is between f(-4) and f(-3), by the Intermediate Value Theorem, there exists a c in the interval (-4 , -3) with the property that f(c) = 0. Such a c is a solution to the equation 3x3 + 9x2 - 3x + 5 = 0.
3) dy dx
= 2 y
2 - y ;
d2y dx2
= 4
(2 - y)3
4) y = -2 x + 3 5) continuous extension exists at origin; f(0) 1.8145 6) 2 7) The roots of f(x) are the solutions to the equation f(x) = 0. Statement (b) is asking for the solution to the equation
4x3 = 3x + 4. Statement (d) is asking for the solution to the equation 4x3 - 3x = 4. These three equations are equivalent to the equations in statements (c) and (e). As five equations are equivalent, their solutions are the same.
8) y = -x + 2 9) c = 16
10)
No, the slope of the graph of y = - tan x is never positive. The slope at any point is equal to the derivative, which is y = - sec2 x. Since sec2 x is never negative, y = - sec2 x is never positive, and the slope of the graph is never positive.
11) 40 m, 20 m/sec
12) f (t) = 2t3 - 6t2 - 2
(2 + t3)2
13) w = - 1 2
z + 15 2
14) 72 m/sec, -32 m/sec2
15) 4 3
16) 0 17) 3
18) 5 3
6
Answer Key Testname: MTH 263 TEST 2 STUDY GUIDE FALL 2020.TMP
19) 20) 0
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