Show all work if full or partial credit is desired. You may use a graphing calculator or Desmos. If Desmos is used, then the device must be in airplane-mode (i.e., no wifi, cellular, or Bluetooth connections). No notes, books or websites allowed.

Part 1 No Calculator

1. `lim_{x to 2} 3=`
2. `lim_{x to 5} x=`
3. `lim_{x to -2} (x^2+3x-4)=`
4. `lim_{x to 0+} 1/x=`
5. `lim_{x to 7} (x^2-49)/(x-7)=`
6. For what values of `x` is `f(x)={(3-(x-1)^2,,x < 1),(x+1,,x >= 1):}` continuous?
   
7. Use the limit definition of the derivative to find the slope of the tangent line to the curve `f(x)=x^2` at `x=-2`.
   1. `f(-2+h)=`
   2. `f(-2+h)-f(4)=`
   3. `(f(-2+h)-f(4))/h=`
   4. `lim_{h to 0}(f(-2+h)-f(-2))/h=`
   5. So, `f'(-2)=`

Part 2 Calculator Allowed

Secants and Tangents

1. Estimate the slope of the tangent line (rate of change) to `f(x)=1/x` at `x=2` by finding the slopes of the secant lines through the points:
   1. `x=2.1` and `1.9`
   2. `x=2.01` and `1.99`
   3. Use the slopes of the secants to estimate the slope of the tangent line accurate to 2 decimal places.
   4. Sketch a graph of `f(x)` on the interval `[0,4]` along with the tangent line when `x=2`.
      

Evaluate the following limits.

2. `lim_{x to 7} (x^2-49)/(x-7)=`
3. `lim_{x to -3} (2x^2+4x-5)=`
4. `lim_{x to 2^-} 1/(x-2)=`
5. `lim_{theta to 0} sin(4theta)/(4theta)=`
6. `lim_{x to 0} (1+2x)^(1/x)=`
7. `lim_{x to 0} (x-9)/(sqrt(x)-3)=`
8. `lim_{x to 0} (sqrt(3x+1)-1)/(2x)=`
9. `lim_{x to 0} (3/(x^2-x)+1/x)=`
10. If `f(x)={((x+3)^2-1,,x < -1),(2x-1,,x >= -1):}`
    1. `lim_{x to -1^-}f(x)=`
    2. `lim_{x to -1^+}f(x)=`
    3. `lim_{x to -1}f(x)=`

Continuity

11. Determine whether `r(x)=(x^2-49)/(x-7)` is continuous at `x=7`. If it is continuous, explain why. Otherwise, explain why it is discontinuous.
12. Determine whether `p(x)=x^2-14x+49` is continuous at `x=7`. If it is continuous, explain why. Otherwise, explain why it is discontinuous.
13. Consider the function `f(x)=1/x` on the interval `-1<=x<=1`.
    `f(-1)=-1` and `f(1)=1`, so, is there a value of `c` in `[-1,1]` such that `f(c)=0`? Why or why not?
14. For what values of `x` is `g(x)=(x-5)/(x+2)` continuous?

The Precise Definition of a Limit

15. State the Precise Definition of a Limit.
16. If `f(x)=2x-1`, `a=-1`, and `epsilon=0.01`, then find an appropriate value of `delta`.
17. If `f(x)=1/2 x+1` and `a=4`, then find `delta` in terms of `epsilon`.

The Derivative at a Point

18. Use the limit definition of the derivative to find the slope of the tangent line to the curve `f(x)=5x^2` at `x=4`.
    Evaluate each of the following and state your answers in simplest form:
    1. `f(4+h)=`
    2. `f(4+h)-f(4)=`
    3. `(f(4+h)-f(4))/h=`
    4. `lim_{h to 0}(f(4+h)-f(4))/h=`
    5. So, `f'(4)=`
19. Use the limit definition of the derivative to find the slope of the tangent line to the curve `f(x)=4x^2` at `x=1`.
20. `f(x)=sqrt(25-x)`. Use the limit definition of the derivative to compute `f'(9)`.

The Derivative Function

21. Use the limit definition of the derivative to find the derivative function when `f(x)=5/x`.
    Evaluate each of the following and state your answers in simplest form:
    1. `f(x+h)=`
    2. `f(x+h)-f(x)=`
    3. `(f(x+h)-f(x))/h=`
    4. `lim_{h to 0}(f(x+h)-f(x))/h=`
    5. So, `f'(x)=`
22. Use the limit definition of the derivative to find the derivative function when `f(x)=3x^2`.
23. `f(x)=sqrt(25-x)`. Use the limit definition of the derivative to find the derivative function `f'(x)`.
24. `f(x)=x/(1-x^2)`. Use the limit definition of the derivative to find the derivative function `f'(x)`.
25. `f(x)=1/x`. Use the limit definition of the derivative to find the second derivative function `f''(x)`.