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The answers to the midterm practice are explained here. Explanations are included. Find the average rate of change of the function over the given interval. Round to nearest thousandth, if necessary. 1) Answer: 22 Explanation: The average rate of change is the change in y divided by the change in x. To find the change in y, calculate the value of y for x = 3 which is y = 54 and at x = -2, y = -56. Subtract 54 - (-56) =110. Divide by the change in x which is 3 - (-2) = 5. The average rate of change is found by using the slope formula from Algebra. Complete the table for the function and find the indicated limit. 2) If f(x) =
Answer: 3.439; 3.940; 3.994; 4.006; 4.060; 4.641 limit = 4.0 Explanation: Use your calculator and enter the function in Y1. Evaluate the functions.for each of the given x values. As you get closer and closer to x = 1 from the right, what is happening to the value of the function? Now as you get closer and closer from the left, what is happening. If the function value is getting closer and closer to one y-value, the limit exists and the y-value is the limit. Use the graph to estimate the limit of the function. 3) Answer: (a) about 2.2, (b) ∞ Explanation: The function value (y-value) is getting closer and closer to 2.2 as x gets closer to3 from the left of 3. . Although the function is defined at x = 3 f(3) =6, the limit limit is 2.2. The value of the limit does not depend on the value of the function. (2.2 is approximate) 4)
Find Answer: 6; -4; does not exist Explanation: As x gets closer to 2 from the left, the function
gets close to 6. As x gets close to 2 from the right, the function gets close to
-4. Evaluate the limit. 5) Answer: Explanation: Direct substitution of x = 4 causes 0 in both the numerator and zero in the denominator. This is an indeterminate form so you must rewrite the expression. Factor and reduce. Evaluate the limit in the reduced fraction since it no longer is undefined at x = 4. 6) Answer: - Explanation: Direct substitution of x = 4 causes 0 in both the numerator and zero in the denominator. This is an indeterminate form so you must rewrite the expression. Factor and reduce. Evaluate the limit in the reduced fraction since it no longer is undefined at x = 2.
7) Answer: Explanation: Direct substitution gives 0 in both numerator and denominator. You can rewrite the expression if you multiply by 1. Use the conjugate of the numerator as the expression to multiply both numerator and denominator. Expand and simplify the numerator but keep the denominator in factored form. Remember you are trying to cancel factors to rewrite.
Write the definition of the derivative. Find the derivative of the given function at the given point using the definition. 8) f(x) = 3 Answer: -6 Explanation: This is a difference quotient and represents the slope of a curve at a point where the x coordinate is equal to a and the y coordinate is f(a). 1) f(a + h) = 3 2) Subtract f(a) = 3 3) Divide by h. The resulting expression will be the difference quotient. Substitute = -1 to find the value. Find the limit, if it exists. 9) Let f(x) =
Answer: Does not exist Explanation: In a piecewise function, evaluate both one-sided limits. The limit from the left is 1, the limit from the right is 6. Since these limits are not equal, the limit does not exist. The graph of this function has a jump at x =3.
Find all points where the function is discontinuous. 10)
Answer: x = -2, x = 0, x = 2 Explanation: This function has discontinuities at x = 2, x = -2 and x = 0. The discontinuity at x = 0 is an infinite discontinuity. The function is not defined at x = -2 and at x = 0. Although the function is defined at x = 2, it is not equal to the limit as x approaches 2.
Find the x-values where a) the function is discontinuous b) the function is not differentiable. 11)
Answer: a) discontinuous at x = 1 b) not differentiable at x = -2, x = 1 Explanation: A function is not differentiable if the slope from the right does not equal the slope from the left. This occurs at an corner or sharp turn. If a function is not continuous it is not differentiable.
Find the intervals on which the function is continuous. 12) y = Answer: (-∞,∞) Explanation: A rational function is not continuous if it is not defined. This occurs when the denominator is 0. Since no real number will evaluate to a zero denominator, the function is defined for all reals.
13) y = Answer: (-∞, 3), ( 3, 8), ( 8, ∞) Explanation: A function is not continuous when it is not defined. A function is not defined when the denominator is 0, so factor the denominator to find the zeros of the denominator. Find the values that cause the denominator to = 0. The function *is* continuous at x = 5. A function is defined as 0 if the numerator is 0 but the denominator is not 0. There is a removable discontinuity at x = 3 and a non-removable one (a vertical asymptote at x = 8) .
Find the equation for the tangent to the curve at the given point. 14) f(x) = 5x2 + x; (-4, 76) Answer: y = -39x - 80 Explanation: First determine the slope. Find the derivative and evaluate it at x = - 4. . Then use the point slope formula.
Solve the problem. 15) Find the points where the graph of the function has horizontal tangents. f(x) = Answer: (- Explanation: A graph has horizontal tangents when its slope (derivative = 0. Calculate the derivative and set it = 0. Solve for x.
16) Find an equation of the tangent to the curve f(x) = Answer: y = 2x - 1 Explanation: Find the derivative function. Set this equal to 2 and solve for x. Find the corresponding y value from f (x) and use the point-slope formula (slope = 2)
Use the definition to find the function's derivative. Then evaluate the derivative at the indicated point. 17) g(x) = 3x2 - 4x, Answer: Explanation: Use
Find the derivative. 18) f(x) = 4x4 + 3x3 + 6 Answer: 16x3 + 9x2 Explanation: Use the power rule on each term. see pages
Find an equation for the line tangent to given curve at the indicated point. 19) y = x - Answer: y = -3x + 4 Explanation: To write the equation, you need the slope and a
point. The point is given and the slope is determined by the derivative
function. To writ the equation of the line, use the point - slope formula.
y -
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds, for the given time interval. Use s(t) to find the indicated quantities. 20) s = - Find the body's velocity and acceleration at the end of the time interval. Answer: The velocity is 6 m/sec. The acceleration is 8 m/ Explanation: When s = position, velocity is
Solve the problem. 21) A rock is thrown vertically upward from the surface of an
airless planet. It reaches a height of Answer: After 30 sec, the rock reaches a maximum height of 1800 m. Explanation: The rock reaches its maximum height when
Find the indicated derivative. 22) Find Answer: - Explanation: Rewrite using negative exponents. y =
Find the derivative. 23) y = Answer: Explanation: Use the quotient rule. Memorize this and be sure to learn this formula in the right order.
24) y = Answer: Explanation: Use the quotient rule. Be sure to memorize this in the correct order.
Solve the problem. 25) Find the tangent to the curve y = Answer: y = -8x + 20 Explanation: Use the quotient rule to find
Find the derivative. 26) y = Answer: Explanation: This is easier if you rewrite using trig identities. Know all derivatives for the 6 functions.
27) r = 18 - Answer:  sin θ
Explanation: The first term in a constant. Its derivative is 0. The second term is a product so use the product rule. Watch the + and - signs.
Find the indicated derivative. 28) Find Answer: 4 Explanation: First find
Find the derivative of the function. 29) y = Answer: Explanation: Differentiate each term separately. Use the chain rule on each term. See page 188-190.
30) r = Answer: Explanation: Use the chain rule. Learn all 6 trig derivatives.
31) f(θ) = sin 
Answer:  sin
+ 2 cos  cos
Explanation: This is a product so use the product rule here. See page 188-190.
32) y = Answer:  (π t - 13) sin
(π t - 13)
Explanation: Rewrite to
33) q = cos Answer: Explanation: Use the chain rule. The outside function is the cos
function and the inside function is
34) h(x) = Answer: Explanation: Use the chain rule. The outside function is a power and the inside function is the quotient. Use the quotient rule to find the derivative of the inside function. Multiply rates.
35) y = Answer: Explanation: Use the chain rule. The power (- 3 )is the outside function. The inside function is 1 + sin 7t. Multiply rates.
Find 36) y = Answer: Explanation: Use the chain rule to find
37) y = 4 cot Answer: Explanation: First find
Find dy/dx by implicit differentiation. 38) x3 + 3x2y + y3 = 8 Answer: - Explanation: Differentiate both sides with respect to x. Don't
forget that each function of y is in turn a function of x so you need the chain
rule. Also, the second term is a product, so use the product rule for that one.
Collect all terms that have
39) y Answer: - Explanation: Use the product rule. Differentiate both sides with
respect to x. Don't forget that each function of y is in turn a function of x so
you need the chain rule. Collect all terms that have
Find the slope at the indicated point on the given curve. 40) Answer: - Explanation: Use implicit differentiation to find
Find the related rate equation. 41) Suppose that the radius r and volume V = 
of a sphere are differentiable functions of t. How is dV/dt related to dr/dt?
Answer:  
Explanation: Related rates. Assume all variable are functions of t = time. Differentiate both sides wrt t. Use the chain rule for the right side.
Solve the problem. 42) A product sells by word of mouth. The company that produces
the product has noticed that revenue from sales is given by Answer: $ 5987/month Explanation: Related rates. The function for revenue is in terms
of x, the u=number of products sold. You are given
43) The radius of a right circular cylinder is increasing at the
rate of Answer: 7π i n.3/sExplanation: Use V = π =2
and  = 9. Find 
when r = 7 and h = 16. Form the related rates equation by differentiating wrt t.
The right side involves a product so you will need to use the product rule.
Substitute given values and find the unknown rate.
Find the absolute extreme values of each function on the interval. 44) F(x) = 3 Answer: Maximum = Explanation: You are given a closed interval on which the function is continuous. This means that there are absolute extreme values. An absolute extreme value can occur at a critical value (where derivative is 0 or not defined). It could also occur at an endpoint. Find the critical l value(s)-the endpoints are given. Make a table of values for each possible value of x. Compare function values to locate the absolute min and absolute max.
Find the extreme values of the function and where they occur. 45) y = Answer: The minimum value is - 2 at x = -1. The maximum value is 2at x = 1. Explanation: Relative extreme values occur when the function
changes direction. This can occur when the derivative is 0 or the derivative
does not exist. Use the quotient rule to find the derivative. Set
Find each critical point and determine if it is a max, min or neither 46) y = Answer:
Explanation: To locate critical points, find the derivative.
Critical points occur when
47) y = Answer:
Explanation: This function is continuous. There is no place the derivative is zero, but the derivative is not defined at x = 1. Use the first derivative test to determine whether is a min or a max or neither.
Use analytic methods to find the intervals on which the function is increasing, decreasing, concave up, and concave down. Also, locate and identify the local extrema and inflection points. 48) y = Answer: increasing at (-∞, - 3) and ( 3, ∞);
decreasing at (- 3, 3); concave down on (-∞, 0); concave up on Explanation: Use the critical numbers from
1) 22
2) 3.439; 3.940; 3.994; 4.006; 4.060; 4.641 limit = 4.0
3) (a) about 2.2, (b) ∞
4) 6; -4; does not exist
5)
6) -
7)
8) -6
9) Does not exist
10) x = -2, x = 0, x = 2
11) a) discontinuous at x = 1 b) not differentiable at x = -2, x = 1
12) (-∞,∞)
13) (-∞, 3), ( 3, 8), ( 8, ∞)
14) y = -39x - 80
15) (-
16) y = 2x - 1
17)
18) 16x3 + 9x2
19) y = -3x + 4
20) The velocity is 6 m/sec. The acceleration is 8 m/
21) After 30 sec, the rock reaches a maximum height of 1800 m.
22) -
23)
24)
25) y = -8x + 20
26)
27)  sin θ
28) 4
29)
30)
31) sin
+ 2 cos cos
32) (π t - 13) sin
(π t - 13)
33)
34)
35)
36)
37)
38) -
39) -
40) -
41) 
42) $ 5987/month
43) 7π in. 3/s
44) Maximum =
45) The minimum value is - 2 at x = -1. The maximum value is 2at x = 1.
46)
47)
48) increasing at (-∞, - 3) and ( 3, ∞); decreasing
at (- 3, 3); concave down on (-∞, 0); concave up on |
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[- 2, 3]
,
find lim f(x) as x approaches 1.
(a) 
f(x)
(b) 
f(x)
f(x) and 
f(x)
and 
f(x)
- 2
for a = -1
-
2 = 
+ 6ah +
.
- 2 from this to get the numerator of the expression. Each term left should have
a factor of h.
f(x)
- 21x
, 14
)
- 2x + 1 that has slope 2.
(3)
(x) = 
.
Evaluate and substitute the given value of x.
at (
2, -2)
=1-2x. At x = 2, the slope is -3.
=m(x =
)
+ 2
- 2t, 0 ≤ t ≤ 2
.
and acceleration is 
.
meters in t seconds. How high does the rock go? How long does it take the rock
to reach its highest point?
if y = 
+ 
.
- 
+
. Use
the power rule on each term.
= 
at the point (1, 12).
+ 
= - 3 csc x cot x + 
= - 4
co
if y = 4 csc x.
x + 4
csc x 
x
= -4 csc x cot
x.. This is a product. So now use the product rule to find 
.
+ 
- 
. Now use the chain rule with the outside function being the power.
= 
sin 
.
Multiply the rates.
(x) = 
cos 7t
.
. This is a
power. To find 
, use the chain
rule. Don't forget to find the derivative of 
in each step. This is easier if you rewrite as 
x.
= 4
= 64, slope at (2, 1)
where x is the number of units produced and sold. If the revenue keeps changing
at a rate of 
per month, how fast is the rate of sales
changing when 1400 units have been made and sold?
when x =
1400. First form the related rates equation containing the two rates of change.
This is done by differentiating both sides of the revenue function wrt t.
Substitute the given values and solve for the unknown rate. See page 208-212
, while the height is decreasing at the rate of 
.
At what rate is the volume of the cylinder changing when the radius is 7 in. and
the height is 16 in.?
-16
+18
,
-1 ≤ x ≤ 4
;
minimum = (3, 27)
=0 or is undefined for a
value in the domain in the original function. Use the product rule to
differentiate. In this problem, rewrite the derivative using positive exponents.
This helps identify where 
- 27x
local maximum: (- 3, 54); local minimum: (3, -54); inflection point at x = 0
to analyze the direction of the function. Use the zeros of 
to analyze the type of concavity of the function. There is a point of inflection
where 
- 
= 
= - 3 csc x cot x + 
= - 4
x + 4 csc x
x
sin 
(x) = 
cot 
