ab calculus rates of change and derivatives review name:ย ยท 5. 2find the x-value(s) where the graph...
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AB Calculus Rates of Change and Derivatives Review Name: _________________________________
1. Find the average rate of change of the function ๐(๐ฅ) = 3 + sin ๐ฅ over the interval [โ๐, ๐].
2. Find the slope of the line tangent to the curve at the given value of ๐ฅ. a) ๐(๐ฅ) = โ3๐ฅ2 + 6๐ฅ at ๐ฅ = 6.
b) ๐(๐ฅ) =
โ1
๐ฅ+6 at ๐ฅ = โ4.
c) ๐(๐ฅ) = 4 โ 15๐ฅ at ๐ฅ = 3.
d) ๐(๐ฅ) = {
8 + ๐ฅ ๐ฅ โค 4โ๐ฅ โ 6 ๐ฅ > 4
at ๐ฅ = 5.
3. Find the equation of the tangent line to the curve ๐(๐ฅ) =
7
๐ฅโ 2 at (3,
1
3).
4. Find the equation of the normal line to the function ๐ฆ = 4๐ฅ2 at (3, 36).
5. Find the x-value(s) where the graph of the function ๐(๐ฅ) = 6๐ฅ2 + 4๐ฅ โ 4 has horizontal tangents.
6. Use the limit definition of derivative to find the derivative of the function ๐(๐ฅ) = ๐ฅ3 + 7.
7. Use the alternative definition of derivative to find the derivative of each function at the indicated point. a) ๐(๐ฅ) = โ๐ฅ at ๐ฅ = 25.
b) ๐(๐ฅ) = โ2๐ฅ2 + 10๐ฅ at ๐ฅ = 10.
8. Consider the graph of f given to the right. a) On what interval is f continuous?
b) On what interval is f differentiable?
9. Graph the given function, then answer the following questions.
๐(๐ฅ) = {3 + ๐ฅ, ๐ฅ โค 0โ2๐ฅ + 3, 0 < ๐ฅ โค 2
๐ฅ2 โ 6๐ฅ + 7, ๐ฅ > 2
a) Compare the right-hand and left-hand derivatives at ๐ฅ = 0 to prove whether or not the function is differentiable at ๐ฅ = 0. Explain your answer.
b) Compare the right-hand and left-hand derivatives at ๐ฅ = 2 to prove whether or not the function is differentiable at ๐ฅ = 2. Explain your answer.
10. For each of the following functions, find the interval for which the function is differentiable. a)
๐(๐ฅ) =1
๐ฅ2 โ 81
b) ๐(๐ฅ) = โ7๐ฅ + 5
c) ๐(๐ฅ) = โ16 โ ๐ฅ2
11. Graph the derivative of the function below on the grid to the right.