[Notes & HW Answers] 3.4 The Chain Rule

[Notes & HW Answers] 3.4 The Chain Rule.pdf

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[Prepwork 3.4]

Q1. The length,L, in micrometers (𝜇μm), of steel depends on the air temperature degrees celsius, and the temperature depends on time,t, measured in hours. If the length of a steel bridge increases by 0.25μm for every degree increase in temperature, and the temperature is increasing at 4 degrees celsius per hour, how fast is the length of the bridge increasing?

A: At a rate of 1 μm/hour 

 

Q2. How many times would we need to use the chain rule to differentiate 𝑓(𝑥)=(1+e^((3+x^2)^1/2))^1/2

A: 3 

 

[HW 3.4]

Q1. Find the derivative of w = (t^3 + 2)^67

A: dw/dt = 201(t^3+2)^66*t^2

Q2. Find the derivative of 𝑓(𝑥)= e^(7x)*(x^2+4^x) 

A: f'(x) = 7e^(7x)*(x^2+4^x) + e^(7x)*(2x + (ln 4)4^x) 

Q3. Find the derivative of v(t) = t^5*e^(-ct). Assume that c is a constant.

A: v'(t) = e^(-ct)*(5*t^4-c*t^5)

Q4. Find the derivative of f(x) = (ax^2 + b)^5. Assume that a and b are constants. 

A: f'(x) = 5(ax^2+b)^4*(2ax)

Q5. Find the derivative of h(z) = (b/(a+z^2)^3. Assume that a and b are constants.

A: h'(z) = 3(b/(a+z^2)^2*(-2bz)/(z^2+a)^2

Q6. Find the derivative of y = (e^(-6t^2) +8)^(1/2). 

A: dy/dt = (1/2)(e^(-6t^2)+8)^(-1/2)*(-12t*e^(-6t^2))

Q7. Use the graph below to find exact values of the indicated derivatives, or state that they do not exist. If a derivative does not exist, enter dne in the answer blank. The graph of 𝑓(𝑥)f(x) is black and has a sharp corner at 𝑥=2x=2. The graph of 𝑔(𝑥)g(x) is blue.

Let h(x) = f(g(x)). Find

A: (A) h'(1) = -3/8

     (B) h'(2) = -3/8

     (C) h'(3) = -3/8

Q8. For what values of 𝑥 is the graph of 𝑦=𝑥𝑒^(−4𝑥) concave down?

A: values = (−∞,1/2​)
(Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10) .