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Worksheet: Applications of Derivatives: Sketching Graphs

Q1:

Determine the local maximum and minimum values of the function 𝑓 ( π‘₯ ) = 1 3 π‘₯ βˆ’ π‘₯ + 1 0 2 .

  • Alocal minimum = 0 , local maximum = 5 2 0
  • Blocal maximum = 0 , local minimum = βˆ’ 5 2 0
  • Clocal maximum = 0 , local minimum = 5 2 0
  • Dlocal minimum = 0 , local maximum = βˆ’ 5 2 0

Q2:

Determine the local maximum and minimum values of the function 𝑓 ( π‘₯ ) = 6 π‘₯ βˆ’ π‘₯ + 4 2 .

  • Alocal minimum = 0 , local maximum = 9 6
  • Blocal maximum = 0 , local minimum = βˆ’ 9 6
  • Clocal maximum = 0 , local minimum = 9 6
  • Dlocal minimum = 0 , local maximum = βˆ’ 9 6

Q3:

Find where (if at all) the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 1 5 π‘₯ + 1 2 has its local maxima and minima.

  • Alocal minimum at π‘₯ = 0 , no local maximum
  • Blocal minimum at π‘₯ = βˆ’ 1 3 , local maximum at π‘₯ = 2 8
  • Clocal maximum at π‘₯ = 1 , no local minimum
  • Dlocal maximum at π‘₯ = βˆ’ 2 , local minimum π‘₯ = 0

Q4:

Find where (if at all) the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 1 5 π‘₯ + 8 2 has its local maxima and minima.

  • Alocal minimum at π‘₯ = 0 , no local maximum
  • Blocal minimum at π‘₯ = 6 , local maximum at π‘₯ = 9
  • Clocal maximum at π‘₯ = 1 , no local minimum
  • Dlocal maximum at π‘₯ = βˆ’ 2 1 , local minimum π‘₯ = 5

Q5:

Find, if they exist, the local maximum and/or minimum values of the function 𝑓 ( π‘₯ ) = 9 𝑒 + 9 𝑒 9 π‘₯ βˆ’ 9 π‘₯ . Also specify what type of value they are.

  • Ahas no local maximum or minimum points
  • B 𝑓 ( 0 ) = 1 8 , local maximum value
  • C 𝑓 ( 0 ) = βˆ’ 1 8 , local maximum value
  • D 𝑓 ( 0 ) = 1 8 , local minimum value
  • E 𝑓 ( 0 ) = βˆ’ 1 8 , local minimum value

Q6:

Find, if they exist, the local maximum and/or minimum values of the function 𝑓 ( π‘₯ ) = 3 𝑒 + 3 𝑒 4 π‘₯ βˆ’ 4 π‘₯ . Also specify what type of value they are.

  • Ahas no local maximum or minimum points
  • B 𝑓 ( 0 ) = 6 , local maximum value
  • C 𝑓 ( 0 ) = βˆ’ 6 , local maximum value
  • D 𝑓 ( 0 ) = 6 , local minimum value
  • E 𝑓 ( 0 ) = βˆ’ 6 , local minimum value

Q7:

For 0 < π‘₯ < 2 πœ‹ 9 find the intervals on which 𝑓 ( π‘₯ ) = 9 π‘₯ βˆ’ 2 9 π‘₯ c o s s i n 2 is increasing or decreasing.

  • A 𝑓 is increasing on the interval ο€Ό πœ‹ 9 , 2 πœ‹ 9  and decreasing on the interval ο€» 0 , πœ‹ 9  .
  • B 𝑓 is increasing on the intervals ο€» 0 , πœ‹ 1 8  and ο€Ό πœ‹ 6 , 2 πœ‹ 9  and decreasing on the interval ο€» πœ‹ 1 8 , πœ‹ 6  .
  • C 𝑓 is increasing on the interval ο€» 0 , πœ‹ 9  and decreasing on the interval ο€Ό πœ‹ 9 , 2 πœ‹ 9  .
  • D 𝑓 is increasing on the interval ο€» πœ‹ 1 8 , πœ‹ 6  and decreasing on the intervals ο€» 0 , πœ‹ 1 8  and ο€Ό πœ‹ 6 , 2 πœ‹ 9  .
  • E 𝑓 is decreasing on the interval ο€Ό πœ‹ 6 , 2 πœ‹ 9  and decreasing on the interval ο€» 0 , πœ‹ 6  .

Q8:

For 0 < π‘₯ < πœ‹ find the intervals on which 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 2 2 π‘₯ c o s s i n 2 is increasing or decreasing.

  • A 𝑓 is increasing on the interval ο€» πœ‹ 2 , πœ‹  and decreasing on the interval ο€» 0 , πœ‹ 2  .
  • B 𝑓 is increasing on the intervals ο€» 0 , πœ‹ 4  and ο€Ό 3 πœ‹ 4 , πœ‹  and decreasing on the interval ο€Ό πœ‹ 4 , 3 πœ‹ 4  .
  • C 𝑓 is increasing on the interval ο€» 0 , πœ‹ 2  and decreasing on the interval ο€» πœ‹ 2 , πœ‹  .
  • D 𝑓 is increasing on the interval ο€Ό πœ‹ 4 , 3 πœ‹ 4  and decreasing on the intervals ο€» 0 , πœ‹ 4  and ο€Ό 3 πœ‹ 4 , πœ‹  .
  • E 𝑓 is decreasing on the interval ο€Ό 3 πœ‹ 4 , πœ‹  and decreasing on the interval ο€Ό 0 , 3 πœ‹ 4  .

Q9:

The graph of the first derivative 𝑓 β€² of a function 𝑓 is shown. On what intervals is 𝑓 concave upward or concave downward?

  • A 𝑓 is concave upward on ( 4 , 6 ) and ( 8 , 9 ) and concave downward on ( 0 , 4 ) and ( 6 , 8 ) .
  • B 𝑓 is concave upward on ( 1 , 2 ) , ( 3 , 5 ) , and ( 7 , 9 ) and concave downward on ( 0 , 1 ) , ( 2 , 3 ) , and ( 5 , 7 ) .
  • C 𝑓 is concave upward on ( 0 , 4 ) and ( 6 , 8 ) and concave downward on ( 4 , 6 ) and ( 8 , 9 ) .
  • D 𝑓 is concave upward on ( 0 , 1 ) , ( 2 , 3 ) , and ( 5 , 7 ) and concave downward on ( 1 , 2 ) , ( 3 , 5 ) , and ( 7 , 9 ) .
  • E 𝑓 is concave upward on ( 4 , 6 ) and ( 8 , 9 ) and concave downward on ( 1 , 4 ) and ( 6 , 8 ) .

Q10:

Determine where 𝑓 ( π‘₯ ) = 3 π‘₯ √ βˆ’ 3 π‘₯ + 1 is concave up and/or where it is concave down.

  • A The function is concave down on the interval ο€Ό βˆ’ ∞ , 2 9  .
  • B The function is concave up on the interval ο€Ό βˆ’ ∞ , 1 3  .
  • C The function is concave up on the interval ο€Ό βˆ’ ∞ , 2 9  .
  • D The function is concave down on the interval ο€Ό βˆ’ ∞ , 1 3  .
  • E The function is concave down on the interval ο€Ό βˆ’ ∞ , 4 9  .

Q11:

Determine where 𝑓 ( π‘₯ ) = 3 π‘₯ √ βˆ’ 4 π‘₯ + 3 is concave up and/or where it is concave down.

  • A The function is concave down on the interval ο€Ό βˆ’ ∞ , 1 2  .
  • B The function is concave up on the interval ο€Ό βˆ’ ∞ , 3 4  .
  • C The function is concave up on the interval ο€Ό βˆ’ ∞ , 1 2  .
  • D The function is concave down on the interval ο€Ό βˆ’ ∞ , 3 4  .
  • E The function is concave down on the interval ( βˆ’ ∞ , 1 ) .

Q12:

Determine the intervals on which is concave up and concave down.

  • A The function is concave up on and concave down on .
  • B The function is concave up on and concave down on .
  • C The function is concave up on and concave down on .
  • D The function is concave up on and concave down on .
  • E The function is concave up on and concave down on .

Q13:

Determine the intervals on which is concave up and concave down.

  • A The function is concave up on and concave down on .
  • B The function is concave up on and concave down on .
  • C The function is concave up on and concave down on .
  • D The function is concave up on and concave down on .
  • E The function is concave up on and concave down on .

Q14:

Let Find the intervals on which the graph is convex downwards and on which it is convex upwards.

  • A convex downwards on , convex upwards on
  • B convex upwards on , convex downwards on
  • C convex downwards on and , convex upwards on
  • Dconvex upwards on and , convex downwards on

Q15:

Let Find the intervals on which the graph is convex downwards and on which it is convex upwards.

  • A convex downwards on , convex upwards on
  • B convex upwards on , convex downwards on
  • C convex downwards on and , convex upwards on
  • Dconvex upwards on and , convex downwards on

Q16:

Determine where the curve defined by π‘₯ = πœƒ s i n and 𝑦 = πœƒ c o s , 0 < πœƒ < 7 πœ‹ 6 , is concave upward and where it is concave downward.

  • A The curve is concave upward on the interval ο€Ό 0 , 7 πœ‹ 6  .
  • B The curve is concave downward on the interval ο€Ό πœ‹ 2 , 7 πœ‹ 6  and upward on the interval ο€» 0 , πœ‹ 2  .
  • C The curve is concave downward on the interval ο€Ό 0 , 7 πœ‹ 6  .
  • D The curve is concave downward on the interval ο€» 0 , πœ‹ 2  and upward on the interval ο€Ό πœ‹ 2 , 7 πœ‹ 6  .
  • E The curve is concave downward on the interval ( 0 , πœ‹ ) and upward on the interval ο€Ό πœ‹ , 7 πœ‹ 6  .

Q17:

The graph of a function 𝑦 = 𝑓 ( π‘₯ ) is shown. At which point is d d 𝑦 π‘₯ negative but d d 2 2 𝑦 π‘₯ positive?

  • Apoint 𝐢
  • Bpoint 𝐡
  • Cpoint 𝐷
  • Dpoint 𝐴
  • Epoint 𝐸