Review the following concepts if needed:

- Rolles Theorem for AP Calculus
- Mean Value Theorem for AP Calculus
- Extreme Value Theorem for AP Calculus
- Test for Increasing and Decreasing Functions for AP Calculus
- First Derivative Test and Second Derivative Test for Relative Extremas for AP Calculus
- Test for Concavity and Points of Inflection for AP Calculus
- Graphing without Calculators for AP Calculus
- Graphing with Calculators for AP Calculus
- Graphs of Derivatives for AP Calculus
- Parametric Curves for AP Calculus
- Polar Equations, Polar Graphs, and Symmetry of Polar Graphs for AP Calculus
- Vectors and Vector Arithmetic for AP Calculus

### Review Problems and Answers

- If
*f*'(*x*) =*x*^{2}– 4, find the intervals where*f*is decreasing. (See Figure 7.6-1.) - If
*f*"(*x*) =2*x*– 6 and*f*' is continuous, find the values of*x*where*f*has a point of inflection. (See Figure 7.6-2.) - See Figure 7.6-3. Find the values of
*x*where*f*has change of concavity. - See Figure 7.6-5. Find the values of
*x*where f has a relative minimum. - See Figure 7.6-7. Given
*f*is twice differentiable, arrange*f*(10),*f '*(10),*f*"(10) from smallest to largest.*Answer*:*f*(10) = 0,*f*'(10) > 0 since*f*is increasing, and*f*"(10) < 0 since*f*is concave downward. Thus the order is*f*"(10),*f*(10),*f*'(10). - See Figure 7.6-8. Find the values of
*x*where*f*' is concave up. - The path of an object is defined by
*x*=2*t*,*y*=*t*+ 1. Find the location of the object when*t*=5. - Identify the shape of each equation: (a) (b) r =6 cos 3θ
- Find the magnitude of the vector and the angle it makes with the positive
*x*-axis.*Answer*: . - If
*a*= {4, – 2},*b*= {– 3, 1}, and*c*= {0, 5}, find 3*a*– 2b +*c*.

*Answer*: Since *f* '(*x*) < 0 if – 2 < *x* < 2, *f* is decreasing on (–2, 2).

*Answer*: Thus, *f* has a point of inflection at *x* = 3.

*Answer*: *f* has a change of concavity at *x* = 0. See Figure 7.6-4.

*Answer*: *f* has a relative minimum at *x* = – 2. See Figure 7.6-6.

*Answer*: *f* ' is concave upward on (– ∞, 0). See Figure 7.6-9.

*Answer*: When *t* =5, *x* =2(5) = 10 and *y* = 5 + 1 = 6 so the location is (10, 6).

*Answer*: (a) spiral, (b) rose

*Answer*: 3 {4, – 2} – 2{– 3, 1} + {0, 5} = {12, – 6} + {6, – 2} + {0, 5} = {18, – 8} + {0, 5} = {18, – 3}.

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