Can you please show work so I can understand the answer. I also need help with the attached assignment

 

1. For some functions f and g, f'(2) = 3 and g'(2) = 5.

Calculate h'(2) when

(a) h(x) = g(x) − f(x)

(b) h(x)=2f(x) + g(x),

(c) h(x) = 11g(x)+1.76.

 

2. Calculate the slope of the tangent line to the curve y = 1/√x²Â at the point (1, 1).

 

3. Calculate the derivative of f(x) = x⁴ + 2x²Â + 5.

 

4. Calculate the slope of the tangent line to the curve y = x³Â + √2x at the point (2, 10) and give an equation for this line.

 

5. For some functions f and g, f(2) = 1, g(2) = 7, f'(2) = 3 and g'(2) = 5.

Calculate h'(2) when

(a) h(x) = g(x) · f(x),

(b) h(x) = g(x)/f(x),

(c) h(x)=3g(x) + f(x)/2.

 

6. Find the points in the x-y plane at which the tangent lines to the curve y = x³Â + 3x²

are horizontal.

 

7. Calculate the derivative of g(x)=(x⁴ + 2x²Â + 5) / (x²Â + 3).

 

8. For some functions f and g, f(1) = 2, f(2) = 1, f'(1) = 7, f'(2) = 3, g(1) = 2, g(2) = 1, g'(1) = 6, and g'(2) = 5. Calculate h'(2) when

(a) h(x) = g(f(x)),

(b) h(x) = f(g(x)).

Calculate h'(1) when

(c) h(x) = g(f(x)),

(d) h(x) = f(g(x)).

 

9. Find the values of x for which the tangent lines to the curve y = x⁴ − x²Â are

horizontal.

 

10. Differentiate f(x) =sqrt(x^3 − 6x + 5) .

 

11. Find the slope of the tangent line to the curve y = √x + x²Â at the point (1, 2).

 

12. The cost to a store of purchasing x cereal boxes is C(x)=1, 000 + x + 0.001x². Each cereal box sells for $4.50 (assume all the purchased boxes do get sold).

(a) Find the revenue and profit as functions of the number of cereal boxes that the store sells.

(b) Calculate the marginal cost, marginal revenue, and marginal profit as functions of the number of bicycles sold.

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