[Notes & HW Answers] 3.1 Powers and Polynomials

[Notes & HW Answers] 3.1 Powers and Polynomials

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[Prepwork 3.1]

Q1. For the derivative of 6x^6+4x^5+1x^4, what is the value of the coefficient of x^5?
A: the coefficient is 36.

Q2. Find the slope of the tangent line to f (x) = x^3 +x^2 +6/x at the point where x = 1.
A: -1.

[HW 3.1]

Q1. Find the derivative of h(q) = 1/(q)^(1/4) .
A: h'(q) = (-1/4)*(1/(q)^(4/5)

Q2. Find the derivative of y = 10t^5−8t^(1/2) + 9/t.
A: dy/dt = 50t^4-4t^(-1/2)-9t^(-2)

Q3. Find the derivative of y = x^(1/2)(x^3+6).
A: dy/dx = 7/2*x^(3/2)+3x^(-1/2)

Q4. Find the derivative of y = (x^7+3)/x.
A: dy/dx = 6x^3-3x^(-2)

Q5. Find the derivative of j(x) = x^8/a + a/b*x^7-cx. Assume that a, b and c are constants.
A: j'(x) = 8/a*x^7+(7a/b)x^6-c

Q6. Find an equation for the line tangent to the graph of f at (2,27), where f is given by f (x) = 4x^3−4x^2+11.
A: y= 32x-37

Q7. On what intervals is the function f (x) = x^8 −8x^3 both decreasing
and concave up?
A: (6,7), (-infinity, 0)

Q8. The height of a sand dune (in centimeters) is represented by f(t) = 600−10t^2 cm, where t is measured in years since 1995. Find f(7) and f'(7), and determine what each means in terms of the sand dune. Give the values of f(7) and f'(7) below, including units.
A: f(7) = 110cm ; f'(7)= -140 cm/yr