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.

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...

Question 1 all parts
a)Compute the derivative dy/dx for y = 5 sqrt (8pi)+ x^6/2 +
26e^x.
b)Compute p(y)' of the function p(y)=4y-5/6y+4
c)write the equation of the tangent line to the graph y=1+9 ln x
at the point that x=1
d)determine the slope of the tangent line to the curve
2x^3+y^3+2xy=13 at the point (1,2)
e)Compute the derivative dw/dz of the function w = (37 + z sin
z)^2

[2 marks] Find the derivative of y =
√
4 sin x + 6
at x = 0.
Consider the following statements. The limit
lim
x→0
g(4 + h) − g(4)
h
is equivalent to:
(i) The derivative of g(x) at x =
h
(ii) The derivative of g(x + 4) at x
= 0
(iii) The derivative of g(−x) at x
= −4
Determine which of the above statements are True (1) or False
(2).
If f (3) = ...

1) Differentiate the function
y= 1/ (9x-8)6
2) Find the derivative of dy/dx of the given function
y= x3(2x-9)7
3) Differentiate the function
y=(4x-4)2(2-x5)2
4) Differentiate the given function
y=(x+4/x-9)8
5) Find the indicated derivative
d/dt ((4t-8)5/t+6)

1. (1’) The position function of a particle is given by s(t) =
3t2 − t3, t ≥ 0.
(a) When does the particle reach a velocity of 0 m/s? Explain the
significance of this value of t.
(b) When does the particle have acceleration 0 m/s2?
2. (1’) Evaluate the limit, if it exists.
lim |x|/x→0 x
3. (1’) Use implicit differentiation to find an equation of the
tangent line to the curve sin(x) + cos(y) = 1
at...

1)Consider the curve y = x + 1/x − 1 .
(a) Find y' .
(b) Use your answer to part (a) to find the points on the curve
y = x + 1/x − 1 where the tangent line is parallel to the line y =
− 1/2 x + 5
2) (a) Consider lim h→0 tan^2 (π/3 + h) − 3/h This limit
represents the derivative, f'(a), of some function f at some number
a. State such an...

a) evaluate the directional derivative of z=F(x,y) = sin(xy) in
the direction of u=(1,-1) at the point (0,pi/2)
b) Determine the slope of the tangent line
c) State the tangent vector

7. For the parametric curve x(t) = 2 − 5 cos(t), y(t) = 1 + 3
sin(t), t ∈ [0, 2π) Part a: (2 points) Give an equation relating x
and y that represents the curve. Part b: (4 points) Find the slope
of the tangent line to the curve when t = π 6 . Part c: (4 points)
State the points (x, y) where the tangent line is horizontal

(a) Find the most general antiderivative of the function f(x) =
−x^ −1 + 5√ x / x 2 −=4 csc^2 x
(b) A particle is moving with the given data, where a(t) is
acceleration, v(t) is velocity and s(t) is position. Find the
position function s(t) of the particle. a(t) = 12t^ 2 − 4, v(0) =
3, s(0) = −1

1. Solve the equation ln(x + 5) − ln(x − 3) = 1 for x.
2.Find all values of x for which the following function has a
tangent line of slope 0 (i.e. f 0 (x) = 0). f(x) = e^2x+1 (2x +
5)
3.Calculate limx→−5 1 − √ x + 6/x + 5

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