Calculus 2.6 Constant, Const.
Multiple, Sum/Difference Notes
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Derivative Rules
Constant: 𝑐 0
Constant Multiple: 𝑐𝑢 𝑐
Sum/Difference: 𝑢 𝑣
Find the derivative of each function.
1. 𝑦 2𝑥 6 2. 𝑦 8√𝑥 2𝜋
Horizontal Tangent Lines
When does a function have a horizontal tangent line? y
The slope of a horizontal tangent line is zero. To find
where a function has a horizontal tangent line, we set the
derivative equal to zero.
x
3. Find the 𝑥-values of any horizontal tangent lines of
𝑓 𝑥 4𝑥 7𝑥 13.
Normal Lines
A normal line goes through the same point the tangent line does, but it is perpendicular to the
tangent line.
4. Find an equation of the NORMAL line of
𝑓 𝑥 𝑥 4𝑥2 𝑥 3 at 𝑥 3.
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Differentiability with piecewise functions
5𝑥 3𝑥 2, 𝑥 1
5. Is the function 𝑓 𝑥 differentiable at 𝑥 1?
7𝑥 3, 𝑥 1
𝑥 𝑎𝑥 2, 𝑥 3
6. What values of 𝑎 and 𝑏 would make the function 𝑓 𝑥
𝑥 𝑏, 𝑥 3
differentiable at 𝑥 3?
2.6 Constant, Constant Multiple, Sum/Difference Rules
Calculus
Practice
Find the derivative of each function.
1. 𝑓 𝑥 2𝑥 4𝑥 5 2. 𝑔 𝑥 5𝑥 𝑥 3. 𝑦 2𝑒 3𝑥 4. 𝑦 𝜋𝑥 𝜋
5. 𝑦 3𝑥 6. ℎ 𝑥 6𝑥 /
4𝑥 /
2 7. 𝑓 𝑥
√
9. 𝑓 𝑥 3𝑥 4𝑥 5𝑥 7 10. 𝑦 4√𝑥 𝑒
8. 𝑓 𝑥 √𝑥 3 √𝑥 2
Find the 𝒙-value(s) where the function has a horizontal tangent.
11. 𝑓 𝑥 4𝑥 12𝑥 13 12. 𝑓 𝑥 𝑥 7 13. 𝑓 𝑥 𝑥
Find the equations of the tangent AND normal lines of each function at the given value of 𝒙.
14. 𝑓 𝑥 3√𝑥 4 at 𝑥 4 15. 𝑦 𝑥 2 at 𝑥 8 16. 𝑓 𝑥 𝑥 2𝑥 2 at 𝑥 2
Tangent: ___________________ Tangent: ____________________
Tangent: ____________________
Normal: ___________________ Normal: ____________________
Normal: ____________________
Are the functions differentiable at the given value of 𝒙?
17. At 𝑥 5. 18. At 𝑥 9. 19. At 𝑥 3.
2𝑥 𝑥 10, 𝑥 5 30 5𝑥 2𝑥 1, 𝑥 3
𝑓 𝑥 𝑥 , 𝑥 9 𝑓 𝑥
50 14𝑥, 𝑥 5 𝑓 𝑥 √𝑥 3𝑥 2𝑥 6, 𝑥 3
𝑥 5𝑥 107, 𝑥 9
What values of 𝒂 and 𝒃 would make the function differentiable at the given value of 𝒙?
20. At 𝑥 1 21. At 𝑥 2. 22. At 𝑥 1.
𝑎 √𝑥 𝑥 2, 𝑥 1 𝑎𝑥 𝑥 4, 𝑥 2 𝑎
𝑓 𝑥 𝑓 𝑥 𝑥 2, 𝑥 1
𝑏𝑥 1, 𝑥 1 𝑏𝑥 5, 𝑥 2 𝑓 𝑥 𝑥
𝑥 𝑏𝑥 1, 𝑥 1
2.6 Constant, Constant Multiple, Sum/Difference Rules Test Prep
23. Given 𝑔 𝑥 2𝑥 where b is a constant, find the value of b if 𝑔′ 2 180.
(A) 10 (B) 20 (C) 40 (D) 80 (E) none of these
24. Calculator required. Which of the following is an equation of the line tangent to the graph of 𝑓 𝑥 𝑥 𝑥
at the point where 𝑓 𝑥 1?
(A) 𝑦 𝑥 1.031
(B) 𝑦 𝑥 0.836
(C) 𝑦 𝑥 0.836
(D) 𝑦 𝑥 0.934
(E) 𝑦 𝑥 1.031
25. lim is
→
(A) 𝑥 (B) 3𝑥 (C) 6𝑥
(D) 6𝑥 (E) nonexistent
ANSWER: C
26. The functions 𝑓 and 𝑔 are given by 𝑓 𝑥 and 𝑔 𝑥 𝑥 2. y
There is a point 𝑃 on the graph of 𝑓 for 𝑥 0 at which the line tangent to
the graph of 𝑓 is perpendicular to the graph of 𝑔. Find the coordinates of
point 𝑃. 2, 1
27.
𝑡
𝑑 𝑡 20𝑡 𝑡 , 0 𝑡 3
6
𝑔 𝑡 , 3 𝑡 16
𝑡 (days) 3 8 12 16
𝑔 𝑡 (cubic feet) 64.5 2100 4050 6500
Mr. Bean is building his own swimming pool by digging up his back yard. For the first three days, he uses a
shovel. After the 3rd day, he uses a backhoe. The amount of dirt that has been removed, in cubic feet, is
modeled by the function 𝑑 defined above, where 𝑔 is a differentiable function and 𝑡 is measured in days.
Values of 𝑔 𝑡 at selected values of 𝑡 are given in the table above.
(a) According to the model 𝑑, what is the average rate of change of the amount of dirt removed over the
time interval 3 𝑡 16 days?
(b) Use the data in the table to approximate 𝑑 10 , the instantaneous rate of change in the amount of dirt
removed, in cubic feet per day, at time 𝑡 10 days. Show the computations that lead to your answer.
(c) Is 𝑑 continuous for 0 𝑡 16? Justify your answer.
(d) Find 𝑑 2 . Use appropriate units.
