1) y = 4- 6 + 8, [- 2, 3]

2) If f(x) = , find lim f(x) x→1

 
6)   
7)  

8) Write the definition of the derivative. Find the derivative of the given function at the given point using the definition. 

12) y =

17. Use the definition to find the function's derivative. Then evaluate the derivative at the indicated point.  g(x) = 3x2 - 4x, (3)

18. Find the derivative. f(x) = 4x4 + 3x3 + 6

19. Find an equation for the line tangent to given curve at the indicated point.   y = x - at ( 2, -2)

23. y =

24 y =
25.Find the tangent to the curve y = at the point (1, 12).

36. Find .

36. y =
37. y = 4 cot
Find dy/dx by implicit differentiation.  

38. x3 + 3x2y + y3 = 8

41) Suppose that the radius r and volume V = π of a sphere are differentiable functions of t. How is dV/dt related to dr/dt?

44. Find the absolute extreme values of each function on the interval. 

F(x) = 3-16+18, -1 ≤ x ≤ 4

1) [- 2, 3]

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.

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.

3) (a) f(x) (b) f(x)

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)

Find f(x) and f(x) and f(x)

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. f(x) does not exist since the two one-sided limits are not equal

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.

8) f(x) = 3 - 2 for a = -1

Answer: -6

3) Divide by h. The resulting expression will be the difference quotient. Substitute = -1 to find the value.

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.

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.

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.

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) .

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.

f(x) = - 21x

Answer: (-, 14), (, - 14)

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) = - 2x + 1 that has slope 2.

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)

17) g(x) = 3x2 - 4x, (3)

Answer: (x) = 6x - 4; (3) = 14

Explanation: Use (x) = . Evaluate and substitute the given value of x.

18) f(x) = 4x4 + 3x3 + 6

Answer: 16x3 + 9x2

Explanation: Use the power rule on each term. see pages

19) y = x - at ( 2, -2)

Answer: y = -3x + 4

To writ the equation of the line, use the point - slope formula. y - =m(x =)

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 and acceleration is .

21) A rock is thrown vertically upward from the surface of an airless planet. It reaches a height of meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point?

Answer: After 30 sec, the rock reaches a maximum height of 1800 m.

Explanation: The rock reaches its maximum height when = 0. Solve for the value of t and substitute it into the function s.

22) Find if y = + .

Answer: - -

Explanation: Rewrite using negative exponents. y = +. Use the power rule on each term.

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.

25) Find the tangent to the curve y = at the point (1, 12).

Answer: y = -8x + 20

Explanation: Use the quotient rule to find . Evaluate at x = 1. Then use the point slope formula.

26) y = +

Answer: = - 3 csc x cot x + x

Explanation: This is easier if you rewrite using trig identities. Know all derivatives for the 6 functions.

27) r = 18 - cos θ

Answer: = - 4cos θ + 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.

28) Find if y = 4 csc x.

Answer: 4 x + 4 csc x x

Explanation: First find = -4 csc x cot x.. This is a product. So now use the product rule to find .

29) y = +

Answer: = -

Explanation: Differentiate each term separately. Use the chain rule on each term. See page 188-190.

30) r =

Answer: = - 2

Explanation: Use the chain rule. Learn all 6 trig derivatives.

31) f(θ) = sin cos

Answer: (θ) = - 11 sin sin + 2 cos cos

Explanation: This is a product so use the product rule here. See page 188-190.

32) y = (π t - 13)

Answer: = - 7π (π t - 13) sin (π t - 13)

Explanation: Rewrite to . Now use the chain rule with the outside function being the power.

33) q = cos

Answer: = sin

Explanation: Use the chain rule. The outside function is the cos function and the inside function is . Multiply the rates.

34) h(x) =

Answer: (x) =

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: = - 21 cos 7t

Explanation: Use the chain rule. The power (- 3 )is the outside function. The inside function is 1 + sin 7t. Multiply rates.

36) y =

Answer:

Explanation: Use the chain rule to find . Simplify. Since is a product, use the product rule to find . There are several ways to express the correct answer.

37) y = 4 cot

Answer: cot

Explanation: First find . 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.

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 as a factor to one side and move the rest of the terms to the other side. Factor .Finally, divide both sides by the multiplier of to solve for .

39) y = 4

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 as a factor to one side and move the rest of the terms to the other side. Factor .Finally, divide both sides by the multiplier of to solve for .

40) = 64, slope at (2, 1)

Answer: -

Explanation: Use implicit differentiation to find . Use the (2, 1) point for (x, y)

41) Suppose that the radius r and volume V = π of a sphere are differentiable functions of t. How is dV/dt related to dr/dt?

Answer: = 4π

Explanation: Related rates. Assume all variable are functions of t = time. Differentiate both sides wrt t. Use the chain rule for the right side.

42) A product sells by word of mouth. The company that produces the product has noticed that revenue from sales is given by 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?

Answer: $ 5987/month

Explanation: Related rates. The function for revenue is in terms of x, the u=number of products sold. You are given = 400. You must find 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

43) The radius of a right circular cylinder is increasing at the rate of , 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.?

Answer: 7π in.3/s

Explanation: Use V = πh since the problem involves the volume of a cylinder. You are given =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.

44) F(x) = 3-16+18, -1 ≤ x ≤ 4

Answer: Maximum = ; minimum = (3, 27)

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.

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 =0 and solve. (Checking for undefined values in the domain of the function is a good habit although that does not affect this problem.) Hint. When you set a fraction = 0, you can multiply both sides by a non-zero denominator. This results in numerator = 0. Solve and determine whether values are minimums or maximums. (Use either first derivative test or second derivative test.)

46) y =

Explanation: To locate critical points, find the derivative. Critical points occur when =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 is not defined. Set = 0 to find locations of horizontal tangents. Since you are working with a fractional equation, multiply by the lcd. For each value you find, tell if it is a min a max or neither.

47) y =

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.

48) y = - 27x

Answer: increasing at (-∞, - 3) and ( 3, ∞); decreasing at (- 3, 3); concave down on (-∞, 0); concave up on local maximum: (- 3, 54); local minimum: (3, -54); inflection point at x = 0

Explanation: Use the critical numbers from 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 = 0 or is undefined and there is a change in concavity.

1) 22

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) (-, 14), (, - 14)

16) y = 2x - 1

17) (x) = 6x - 4; (3) = 14

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) = - 3 csc x cot x + x

27) = - 4cos θ + sin θ

28) 4 x + 4 csc x x

29) = -

30) = - 2

31) (θ) = - 11 sin sin + 2 cos cos

32) = - 7π (π t - 13) sin (π t - 13)

33) = sin

34) (x) =

35) = - 21 cos 7t

36)

37) cot

38) -

39) -

40) -

41) = 4π

42) $ 5987/month

43) 7π in.3/s

44) Maximum = ; minimum = (3, 27)

45) The minimum value is - 2 at x = -1. The maximum value is 2at x = 1.

48) increasing at (-∞, - 3) and ( 3, ∞); decreasing at (- 3, 3); concave down on (-∞, 0); concave up on local maximum: (- 3, 54); local minimum: (3, -54); inflection point at x = 0

Farber            Practice For Midterm test in Calculus I

Find the average rate of change of the function over the given interval.  Round to nearest thousandth, if necessary.

1)  y = 4 x³- 6 x²  + 8, in the closed interval [- 2, 3]

Complete the table for the function and find the indicated limit.  

                                                                                                                   find  lim f(x) x -> 1

3)Use the graph to estimate the limit of the function.

  (a) lim x->3    (b) lim x -> 0⁺  

4) 

Find 

Find and and     

Evaluate the limit.

5)     

6)      

7)   

Write the definition of the derivative. Find the derivative of the given function at the given point using the definition.

8)  f(x) =  3 x²  - 2  for a =  -1

Find the limit, if it exists.

9) 

f(x)

Find all points where the function is discontinuous.

10) 

Find the x-values where a) the function is discontinuous b) the function is not differentiable.

11) 

Find the intervals on which the function is continuous.

12)  y =  

13)  y =  

Find the equation for the tangent to the curve at the given point.

14)  f(x) = 5x² + x; (-4, 76)

Solve the problem.

15)  Find the points where the graph of the function have horizontal tangents.

f(x) = x³ -  21x  

16)  Find an equation of the tangent to the curve f(x) = x² - 2x + 1 that has slope 2.

Use the definition to find the function's derivative.  Then evaluate the derivative at the indicated point.

17)  g(x) = 3x² - 4x, 

Find the derivative.

18)  f(x) =  4x⁴  + 3x³  + 6

Find an equation for the line tangent to given curve at the indicated point.

19)  y = x - x² at ( 2,  -2)

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 = - t³ +  2t² -  2t, 0 ≤ t ≤  2

Find the body's velocity and acceleration at the end of the time interval.

Solve the problem.

21)  A rock is thrown vertically upward from the surface of an airless planet.  It reaches a height of    meters in t seconds.  How high does the rock go?  How long does it take the rock to reach its highest point?

Find the indicated derivative.  

22) Find if y = + .

23) y =

24) y =

25) Find the tangent to the curve y = at the point (1, 12).

Find the derivative.

26)  y =  +  

27)  r =  18 - cos θ

Find the indicated derivative.

28)  Find  if y =  4 csc x.

Find the derivative of the function.

29)  y =  +  

30)  r =  

31)  f(θ) = sin  cos  

32)  y = cos ⁷ (π t -  13)

33)  q = cos  

34)  h(x) =  

35)  y =  

Find .

36)  y =  

37)  y =  4 cot  

Find dy/dx by implicit differentiation.

38)  x³ + 3x²y + y³ = 8

39)  ^y = 4

Find the slope  at the indicated point on the given curve.

40)   =  64, slope at (2, 1)

Find the related rate equation.

41)  Suppose that the radius r and volume V = 4/3π r³ of a sphere are differentiable functions of t.  How is dV/dt related to dr/dt?

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  where x is the number of units produced and sold. If the revenue keeps changing at a rate of $400 per month, how fast is the rate of sales changing when  1400 units have been made and sold?  

43)  The radius of a right circular cylinder is increasing at the rate of 2 in/sec, while the height is decreasing at the rate of 9 in/sec. At what rate is the volume of the cylinder changing when the radius is 7 in. and the height is 16 in.?

Find the absolute extreme values of each function on the interval.

44)  F(x) = 3 x⁴-16 x³ +18x²,  -1 ≤ x ≤ 4

Find the extreme values of the function and where they occur.

45)  y =  

Find  each critical point and determine if it is a max, min or neither

46)  y =  

47)  y =  

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 =  -  27x

1)  22  

2)  3.439; 3.940; 3.994; 4.006; 4.060; 4.641

or evaluate by algebraic techniques.

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)  (- ,  14 ), ( , - 14 )  

16)  y = 2x - 1  

17)  (x) = 6x - 4; (3) = 14  

18)  16x³  + 9x²    

19)  y =  -3x +  4  

20)  The  velocity is  6 m/sec.  The acceleration is  8 m/ sec².  

21)  After  30 sec, the rock reaches a maximum height of  1800 m.  

22)  -  -    

23)   =    

24)   =    

25)  y =  -8x +  20  

26)   = -  3 csc x cot x + sec²x  

27)   = -  4 cos θ + sin θ  

28)  4 csc³x +  4 csc x cot²x  

29)   =   -      

30)   = - ²    

31)  (θ) = -  11 sin  sin  +  2 cos  cos    

32)   = -  7π cox⁶(π t -  13) sin (π t -  13)  

33)   =  sin    

34)   (x) =    

35)   = -  ²¹ cos  7t  

36)      

37)     cot    

38)  -    

39)  -    

40)  -    

41)   = 4π  

42)  $ 5987/month

43)  7π in.³/s

44)  Maximum = (-1 , 37); minimum = (3, 27)  

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   local maximum: (-  3,   54); local minimum: (3,   -54); inflection point at x = 0

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