NJCTL.org DERIVATIVES UNIT PROBLEM SETS PROBLEM SET #1 – Rate of Change ***Calculators Not Allowed*** Find the average rate of change for each function between the given values: 1. = 2 + 4 + 3 from = −2 to =3 2. () = 3 − 2 +1 from = −2 to =2 3. () = from = 2 to = 4. ℎ() = 2 from = 4 to = 3 4 5. () = √ 2 −9 from = −5 and =3 6. = log 10 from = 10 to = 100
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DERIVATIVES UNIT PROBLEM SETS PROBLEM SET #1 ...content.njctl.org/courses/math/ap-calculus-ab/...2014/12/07 · PROBLEM SET #12 – Contin. vs. Differentiability ***Calculators Not
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DERIVATIVES UNIT PROBLEM SETS
PROBLEM SET #1 – Rate of Change ***Calculators Not Allowed***
Find the average rate of change for each function between the given values:
1. 𝑦 = 𝑥2 + 4𝑥 + 3 from 𝑥 = −2 to 𝑥 = 3
2. 𝑓(𝑥) = 𝑥3 − 𝑥2 + 1 from 𝑥 = −2 to 𝑥 = 2
3. 𝑔(𝑥) = 𝑠𝑖𝑛𝑥 from 𝑥 =𝜋
2 to 𝑥 = 𝜋
4. ℎ(𝑥) = 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 from 𝑥 =𝜋
4 to 𝑥 =
3𝜋
4
5. 𝑟(𝑡) = √𝑡2 − 9 from 𝑡 = −5 and 𝑡 = 3
6. 𝑦 = log10 𝑡 from 𝑡 = 10 to 𝑡 = 100
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7. 𝑦 = 2 log5 𝑡 from 𝑡 = 1 to 𝑡 = 25
8. 𝑔(𝑡) = 𝑒𝑡 + 5 from 𝑡 = 0 and 𝑡 = 1
9. 𝑦 = |4 − 𝑥2| from 𝑥 = 0 to 𝑥 = 3
10. 𝑓(𝑥) =𝑥+2
𝑥−2 from 𝑥 = 3 to 𝑥 = 8
11. Andrew is a physics student testing the rate of change of objects he can throw. Given his calculations, if he throws the baseball from the top of a hill, it follows the equation 𝑥(𝑡) = −4.9𝑡2 + 14.7𝑡 + 25. He wants to know the average rate of change of the ball for each of the following time periods: (Calculator allowed) a) 𝑡 = 0 𝑡𝑜 𝑡 = 1.5 b) 𝑡 = 1.5 𝑡𝑜 𝑡 = 3 c) 𝑡 = 1 𝑡𝑜 𝑡 = 3
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PROBLEM SET #2 – Slope of a Curve ***Calculators Not Allowed***
For problems #1-8, find the limit of the function at the given point:
1. a) Find the derivative of the function 𝑦 = 4𝑥 + 1.
b) What is the value of the derivative at 𝑥 = 1 ?
2. a) Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) = 2𝑥2. b) What is the value at 𝑥 = 2 ?
3. a) Find the derivative of 𝑔(𝑥) = 3𝑥3 + 1.
b) What is the value of 𝑔′(1) ?
4. a) Find 𝑑𝑦
𝑑𝑥 of the function 𝑦 = (𝑥 + 1)2
b) What is the slope at 𝑥 = −1 ?
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5. a) Find ℎ′(𝑥) if ℎ(𝑥) = 2𝑥3 + 3𝑥2. b) What is the slope at 𝑥 = −2 ?
6. a) Find 𝑑𝑓
𝑑𝑟 if 𝑓(𝑟) = 𝜋𝑟2.
b) What is the slope at 𝑟 = 3 ?
7. a) Find the derivative of 𝑦 = √𝑥.
b) What is the slope at 𝑥 = 2 ?
8. Of the functions you have worked with, which type of functions have the same
average rate of change as their instantaneous rate of change? (i.e. Δ𝑦
Δ𝑥= 𝑦′) Try taking
the average rate of changes of the examples above. a) Linear b) Quadratic c) Cubic d) Square Root
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PROBLEM SET #3 – Derivative Rules ***Calculators Not Allowed***
1. Find the derivative of the function 𝑦 = 11.
2. Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) = 4𝑥2.
3. a) Find the derivative of 𝑔(𝑥) = 12𝑥3 − 4𝑥2 + 2𝑥.
b) What is the value of 𝑔′(−1) ?
4. Find 𝑑𝑦
𝑑𝑥 for the function 𝑦 = (𝑥 + 2)2.
5. a) Find ℎ′(𝑥) if ℎ(𝑥) = 2𝑥3/4.
b) What is the slope at 𝑥 = 16 ?
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6. a) Find 𝑑𝑓
𝑑𝑡 if 𝑓(𝑡) = 3√𝑡 .
b) What is the slope at 𝑡 = 2 ?
7. a) Find the derivative of 𝑦 = (2𝑥 + 1)(2𝑥 − 1) .
b) What is the slope at 𝑥 = 2 ?
8. a) Find 𝑔′(𝑥) if 𝑔(𝑥) = (𝑥2 + 4𝑥 + 1)2 .
b) What is the slope at 𝑥 = 0 ?
9. a) Find 𝑑𝑦
𝑑𝑥 if 𝑦 =
1
2√𝑥3
.
b) What is the slope at 𝑥 = 8 ?
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PROBLEM SET #4 – Higher Order Derivatives ***Calculators Not Allowed***
1. Find the 1st and 2nd derivative of the function 𝑦 = 20𝑥 .
2. Find 𝑓 ′(𝑥) and 𝑓 ′′(𝑥) of the function 𝑓(𝑥) = 13𝑥2 + 4𝑥 .
3. Find the 1st and 2nd derivative of 𝑔(𝑥) = (𝑥 + 4)3 .
4. Find 𝑑𝑦
𝑑𝑥 and
𝑑2𝑦
𝑑𝑥2 of the function 𝑦 = (𝑥 − 4)2 .
5. Find ℎ′′′(𝑥) if ℎ(𝑥) =1
24𝑥4 −
1
6𝑥3
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6. Find 𝑑2𝑓
𝑑𝑡2 if 𝑓(𝑡) = √𝑡 .
7. Find the derivative of 𝑦 = √𝑥3
.
8. Find the 1st and 2nd derivatives of 𝑓(𝑥) = 𝑥𝑏 a) where 𝑏 = 2 b) where 𝑏 = 3 c) generically for all 𝑏 > 3
9. Looking back at problem 8c above, we begin to see a pattern with derivatives. As we take derivatives, our exponents are used as coefficients and multiplied together. If we were to continue taking derivatives until the exponent is zero, we can see that the last coefficient is equal to the factorial (!) of the original power (any derivative after that will equal zero). If we stopped at any other time along that path, we would have a permutation of the exponents equal to 𝑎𝑃𝑑 where “a” is the original exponent and “d” is the specific derivative. Using this method, find the following derivatives: (calculator OK)
a) 8th derivative of 𝑥9 b) 10th derivative of 2𝑥10
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PROBLEM SET #5 – Trigonometry Rules ***Calculators Not Allowed***
1. a) Find the derivative of the function 𝑦 = 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 .
b) What is the value of the derivative at 𝑥 =𝜋
2 ?
c) What is the value of the derivative at 𝑥 =𝜋
6 ?
2. a) Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) =1
𝑠𝑖𝑛𝑥+
1
𝑐𝑜𝑠𝑥 .
b) What is the value of the derivative at 𝜋
4 ?
c) What is the value of the derivative at 𝜋
3?
3. a) Find the derivative of 𝑔(𝑥) = 3𝑡𝑎𝑛𝑥 − 4𝑐𝑜𝑡𝑥 .
b) What is the value of 𝑔′ (𝜋
6) ?
4. a) Find 𝑑𝑦
𝑑𝑥 of the function 𝑦 =
1
𝑐𝑜𝑡𝑥+
2
𝑐𝑜𝑡𝑥 .
b) What is the derivative at 𝑥 =𝜋
4 ?
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5. a) Find ℎ′(𝑥) if ℎ(𝑥) = −𝑐𝑠𝑐𝑥 + 𝑠𝑒𝑐𝑥 .
b) What is the slope at 𝑥 =5𝜋
6 ?
6. Find the first four derivatives of 𝑦 = 4𝑠𝑖𝑛𝑥 .
7. Find the first four derivatives of 𝑔(𝑥) = 5𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥
8. Find 𝑑𝑚
𝑑𝑥 if 𝑚(𝑥) = tan−1 𝑥 + cot−1 𝑥 .
9. a) Find the derivative of 𝑦 = 2 sin−1 𝑥 .
b) What is the value of the derivative at 𝑥 = 0.
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PROBLEM SET #6 – Product/Quotient Rule ***Calculators Not Allowed***
1. Find 𝑓 ′(𝑥) if the function is 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2) .
2. Find 𝑑𝑦
𝑑𝑥 of the function is 𝑦 = (𝑥2 + 4𝑥 − 3)(3𝑥2 − 10) .
3. Find the derivative of 𝑔(𝑥) = √𝑥(𝑥4 − 3𝑥2 − 10𝑥 + 1) .
4. Find ℎ′(𝑥) if ℎ(𝑥) = √𝑥3
𝑡𝑎𝑛𝑥 .
5. Find 𝑦′ if 𝑦 = 𝑐𝑜𝑠𝑥 ∙ 𝑠𝑖𝑛𝑥 .
6. Find 𝑓 ′(𝑥) if𝑓(𝑥) = sin2 𝑥 .
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7. Find the derivative of 𝑦 =𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥+
𝑐𝑜𝑠𝑥
𝑠𝑖𝑛𝑥 (without using trig shortcuts).
8. Find 𝑔′(𝑥) if 𝑔(𝑥) =𝑥2+1
𝑥2−1 .
9. Find 𝑑𝑦
𝑑𝑡 if 𝑦 =
𝑒𝑡+1
𝑡3 .
10. Using any prior rules, find the 1st and 2nd derivatives of 𝑦 = 𝑡𝑎𝑛𝑥 .
11. **Show using product rule that the derivative of sin3 𝑥 is 3 sin2 𝑥𝑐𝑜𝑠𝑥 .
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PROBLEM SET #7 Derivatives Using Tables ***Calculators Allowed***
x f(x) f’(x) g(x) g’(x)
-2 -19 16 -11 19
-1 -6 10 -2 4
0 1 4 -1 -1
1 2 -2 -2 6
2 -3 -8 1 7
1. Given ℎ(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥) find: ℎ′(2)
find: ℎ′(1)
2. Given 𝑗(𝑥) = 2𝑥2 ∙ 𝑓(𝑥) find: 𝑗′(0)
find: 𝑗′(2)
3. Given 𝑘(𝑥) =𝑓(𝑥)
𝑔(𝑥) find: 𝑘′(−1)
find: 𝑘′(1)
4. Given 𝑚(𝑥) = 𝑔(𝑥)2 find: 𝑚′(0)
find: 𝑚′(−2)
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t g(t) g’(t) h(t) h’(t)
0 -10 6 -1 -9
1 -3 1 -3 5
2 4 14 -1 -1/4
3 2 -1.5 -2 5
4 0 8 1/2 1
5. Given 𝑓(𝑡) = 𝑔(𝑡) ∙ ℎ(𝑡) find: 𝑓 ′(1)
find: 𝑓 ′(2)
6. Given 𝑟(𝑡) =𝑔(𝑡)
ℎ(𝑡) find: 𝑟 ′(0)
find: 𝑟 ′(3)
7. Given 𝑠(𝑡) = ℎ(𝑡)2 find: 𝑠′(0)
find: 𝑠′(4)
8. Given 𝑔(𝑡) = 2𝑡3ℎ(𝑡) find: 𝑔′(1)
find: 𝑔′(2)
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PROBLEM SET #8 – Tangent/Normal Lines ***Calculators Not Allowed***
For each question find the equations of the tangent & normal lines at the given value.
1. 𝑓(𝑥) = 4𝑥 + 7 at 𝑥 = 2
2. 𝑓(𝑥) = 3𝑥2 − 2𝑥 + 1 at 𝑥 = −3
3. 𝑦 = 3𝑐𝑜𝑠𝑥 + 1 at 𝑥 =𝜋
2
4. 𝑔(𝑥) = −4√𝑥 − 2 at 𝑥 = 4
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5. 𝑦 = 𝑐𝑜𝑠𝑥 ∙ 𝑠𝑖𝑛𝑥 at 𝑥 =𝜋
4
6. 𝑦 = |2𝑥| at 𝑥 = −3
7. ℎ(𝑥) =𝑥2+1
𝑥2−1 at 𝑥 = 0
8. 𝑓(𝑥) =1
𝑥 at 𝑥 = 2
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PROBLEM SET #9 – Derivatives of Log & e ***Calculators Not Allowed***