Question

1. (5pts.) Compute the derivative dy/dx for y = 7√ 9π + x ^5 /6 + 27e^x .

3. (5pts.) Write the equation of the tangent line to the graph of y = 3 + 8 ln x at the point where x = 1.

4. (5pts.) Determine the slope of the tangent line to the curve 2x^3 + y^3 + 2xy = 14 at the point (1, 2).

5. (5pts.) Compute the derivative dw/dz of the function w = (33 + z sin z)^6 .

6. (5pts.) Find the limit: lim θ→0 (sin(6θ) cos(6θ)) / 5θ =

7. (5pts.) Find the limit: limx→3 (x ^2 − 7x + 12) / (x^ 2 − 9) =

8. (5pts.) Find the limit: limx→∞ (49x^ 4 + 51) / (6x^ 8 + 11) =

10. (6pts.) Show that the derivative of f(x) = 1 + 8x^ 2 is f ‘(x) = 16x by using the definition of the derivative as the limit of a difference quotient.

11. (5pts.) If the area A = s^ 2 of an expanding square is increasing at the constant rate of 4 square inches per second, how fast is the length s of the sides increasing when the area is 16 square inches?

12. (5pts.) Find the intervals where the graph of y = x ^3 − 5x^ 2 + 2x + 4 is concave up and concave down, and find all the inflection points.

14. (6pts.) Find the absolute maximum and minimum values of f(x) = x ^3 −3x on the closed interval [0, 3].

15. (6pts.) A particle moves along the x-axis with an acceleration given by a(t) = 6t + 2, where t is measured in seconds and s (position) is measured in meters. If the initial position is given by s(0) = 3 and the initial velocity is given by v(0) = 1 then find the position of the particle at t seconds.

18. (5pts.) Find the area under the curve y = 2 + 2e^ x from x = 0 to x = 1

Answer 1

Sorry for not solving all the questions as it is advised here to solve only 1st question in case of multiple questions.

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