Sample Mathematics Answers

1. The graph is as below, and the slope is -2.

2. A) By the power rule, the derivative is 2 * 1/4 * x^2-1, or 1/2*x¹, or just x/2.
B) By the constant rule, the derivative is 0.
C) By the linear rule, the derivative is -2. Note that this is the same as the slope that you calculated in problem 1.
D) By the power rule, the derivative is 3 * 4/3 * x^4/3–1, or just 3x^1/3.

3. When q₁=1, the equation reduces to p = 7 - q₂; when q₁=3, the equation reduces to
p = 11 - q₂. The graphs are shown below. The increase in q₁ causes the curve to shift up.

4. The partial derivative with respect to q₁ is just the derivative when q₂ is treated as a constant; it is 2. The partial derivative with respect to q2 is –2q².

5. The slope of the curve is given as 1/2; the change in x between 5 and 7 is 2. Therefore the change in y should be 1/2 * 2 or 1. Thus the new value of y should be about 5.

6. The derivative of x³ is 3*x². When x=2, the slope of the tangent line is 3*2² = 3*4 = 12. If x rises to 3.1, then the rise in y should be approximately equal to 12*0.1, or 1.2. Since 2³=8, we would expect y to be about equal to 8+1.2 = 9.2 when x=2.1. A calculator shows that 2.1³ = 9.261, so it's pretty good.

7. If Q = 5 + P and Q = 17 - 2P, then 5 + P = 17 - 2P. Adding 2P to both sides and subtracting 5 from both sides gives 3P = 12, or P = 4. If P = 4, then you can use either of the original equations to find Q = 5 + 4 = 9, or Q = 17 - 2*4 = 9. (If they didn't give the same answer you would know you had made a mistake.)