1. Find the derivative of the following expressions. DO NOT SIMPLIFY YOUR ANSWERS (2 pts each = 12 pts)
   6. f(x)=(x+4)(2-x)4
2. A balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high? (5 pts)
3. If $12,000 is invested in an account that earns 3.95% compounded continuously, find theinstantaneous rate of change of the amount in the account when the account is worth $25,000. No credit without work shown. (8 pts)
4. Use the four-step process to find the derivative off(x)= -x2+6x-3 . No credit for answers without using the four-step process. (5 pts)
5. The price-demand equation for hamburgers at a fast-food restaurant isx+400p=3000. Currently the price of a hamburger is $4. Would you recommend the restaurant increase the price of the hamburger by 10%? Explain. Use elasticity of demand to provide justification for your answer. (5 pts)
6. Use the information given toneatly sketch the graph off. Construct sign charts forf' andf''and explain what they tell you about the graph off. Include labels of scale and important points on the graph. You may assume thatfis continuous on its domain and that all intercepts are included in the information given: (5 points)

• Domain is all real numbers
• f(-2)=1, f(0)=0, f(2)=1
• f'(0)=0
• f'(x)<0on(-,0)
• f'(x)>0on(0,)
• f''(-2)=0, f''(2)=0
• f''(x)<0on(-,-2) and(2,)
• f''(x)>0 on(-2,2)
• x-f(x)=2
• xf(x)=2

1. Sketch the graph off(x)=x2. (8 pts total)
   1. Carefully draw thesecantline to the graph off passing through the points (1,1) and (3,9). Use algebra to find the slope of this line and its equation. Is this slope an average rate of change or instantaneous rate of change? (3 pts)
   2. Carefully draw thetangentline to the graph off atx=1. Use calculus to find the slope of this line and its equation. Is this slope an average rate of change or instantaneous rate of change? (3 pts)
   3. Use the following vocabulary from our course to explain how a secant line turns into a tangent line. Use each at least once. (2 pts)
      1. Average rate of change
      2. Instantaneous rate of change
      3. Secant line
      4. Tangent line
      5. Derivative
      6. Slope
      7. Limit
      8. Difference quotient
2. A company manufactures and sellsx solar panels per month. The monthly cost and price-demand equations are, respectively, (22 pts total)

C(x)=200x+100,000 x+80p=24,000 for0

1. Write revenue as a function of demand. (2 pts)
2. Use calculus to find the maximum revenue. (2 pts)
3. Write profit as a function of demand (2 pts)
4. Use calculus to determine how many solar panels the company should manufacture each month to maximize its profit. (2 pts)
5. What is this maximum monthly profit? (2 pts)
6. How much should the company charge for each solar panel to maximize profit? (2 pts)
7. Write the marginal profit function and explain what it means at a production level of 2000 solar panels. (2 pts)
8. If the company receives a tax benefit of $10 for each solar panel it produces, how many solar panels should the company manufacture each month to maximize its profit? Note: this tax benefit adds $10 per solar panel produced to the profit. (2 pts)
9. With the tax benefit, now what is the maximum monthly profit? (2 pts)
10. With the tax benefit, now how much should the company charge for each solar panel? (2 pts)
11. With the tax benefit, write the new marginal profit function and explain what it means at a production level of 2000 solar panels. (2 pts)