MATH 1211 - Calculus - Study for Exam

Graph the function f(x) = x^2 - 2 in the interval [0,8] showing partition into 4 subintervals of equal length and estimating the area under the function using mid-point sum Σ4_k=1 f(c_k) △x_k

(Goes to 8, then up to 24, then up to 48)

Suppose a snowball melts and its surface area decreases at the rate of 1 cm^2/min, find the rate at which the diameter decreases when it reaches 10 cm. [Hint: surface area of a sphere is 4pir^2].

-1/20pi

What is the derivative of f(x) = 6x^-2 + 8x^3 + 11x + e^x?

-12x^-3 + 24x^2 + 11 + e^x

Suppose that f and g continuous and that ∫10_6 f(x)dx=-3 and ∫10_6 g(x)dx=9. Find ∫10_6 [4f(x)+g(x)]dx

-3

If xy^2 + 2xy = 8 then at point (1,2) y' is

-4/3

Evaluate the lim(x->-2) (x^2-x-6)/(x-2)

0

Let f(x) = x^2 + 3. Find lim h->0 (f(x+h)-f(x))/h

0

Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for f(x) = x^2 - 2x - 8 on [1, 3].

1

The sides of the rectangle change according to dz/dt = 1, dx/dt = 3 dy/dt. What is dx/dt at the instant whenx = 4 and y = 3? [Hint: z^2 = x^2 + y^2]

1

What is lim x->0 (e^x-1)/tan(x)

1

Estimate the area under the graph f(x) = x^2 between x=0 and x=2 using left sum with two rectangles of equal width.

1.0

Find the maximum value of f(x) = x - 2lnx on the interval [1, 4].

1.2274

∫0_1 x/(x^2+9)

1/2(ln10 - ln9)

Suppose we know f(x) that is continuous and differentiable on the interval [-7,0], that f(-7) = -3 and that f'(x) <= 2. What is the largest possible value for f(0)?

11

If f(t) = t^2 + t , Between t = 10 and t = 12 , what is the increase in f divided by the increase in t

23

The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20π meters?

4π m^2/sec

Find the minimum cost of a rectangular box that is three times as long as it is wide and which holds 100 cubic centimeters of water. The material for the bottom costs 7¢ per cm^2, the sides cost 5¢ per cm^2 and the top costs 2¢ per cm^2

About $6.87

Below is the graph of a function y = f(x). Identify all the relative extrema and absolute extrema of the function. (Passes through (-1,2), then down to (2, -6) and then up again)

Absolute Max (4,5); Absolute Min (2,-6); Relative Max's (-1,2) and (4,5); Relative Mini's (-3,-2) and (2,-6)

Note the function below. Find all of the relative extrema and absolute extrema of the function. (Goes through (-2, -1), then up to (1, 3) then down to (2, -4), then up to (6, 8). and then down)

Absolute Max (6,8); Absolute Min (9,-6); Relative Max's (1,3) and (6,8); Relative Mini's (-2,-1) and (2,-4)

For which of the following does lim(x->3) f(x) exist?

I and II only

The graph of a function is given. Choose the answer that represents the graph of its derivative. (Looks like an M)

d. The one that passes through (-5,0) and (5,0). Goes from top to bottom.

The graph of a function is given. Choose the answer that represents the graph of its derivative. (Looks like it goes down and the up symmetrically)

d. bottom of x axis, and then at (0,0) switches to top of x axis

Let y = x^x find the value of dy/dx at x=3.

dy/dx = 27ln3 + 27

Find the derivative of y = (cotx)/(2x-5)

dy/dx = [-(2x-5)csc^2x-2cotx]/(2x-5)^2

Use product rule to find the derivative of f(x) = (x^2 + 5cosx) (x^3 - 2x + tanx).

f'(x) = (2x - 5sinx)((x^3 - 2x + tanx) + (x^2 + 5cosx)(3x^2 + sec^2x - 2)

Find f(x) for f(x) = (x^2+1)^3

f'(x) = 6x(x^2+1)^2

Find f'(x) for f(x) = sec8x

f'(x) = 8sec8xtan8x

If f(θ) = 4^(θ-2)/sin(πθ), then find f'(θ)

f'(θ) = (4^(θ-2)[(ln4)sin(πθ)-πcos(πθ)])/sin^2(πθ)

Let f(x) = x^(-9), find the antiderivative

f(x) = (-1/8)^(-8) + C

Find the antiderivative of f(x) = 3/x

f(x) = 3lnx + C

Find the linear function with f(0) = 3 and slope 6.

f(x) = 6x + 3

Which function has the same derivative function f(x) as g(x) = ln + x^2 - x?

f(x) = lnx + x^2 - x + 10

Which function has the same derivative function f(x) as g(x) = x^2 + lnx?

f(x) = x^2 + lnx + 3

Suppose g(x) is a polynomial function such that g(-1) = 4 and g(2) = 7. Then there is a number c between -1 and 2 such that

g'(c) = 1

Find the derivative of g(x) = (4)^(3x^3)

g'(x) = (4)^(3x^3) (9x^2) (ln4)

Use quotient rule to find the derivative of g(x) = x^2/(x-11)

g'(x) = (x^2 - 22x)/(x-11)^2

The function f is continuous on the closed interval [0, 2] and that has values given in the table. The equation f(x) = 1/2 must have at least two solutions in the interval [0, 2], if k = ? x 0 1 2 f(x) 1 k 2

k = 0

The graph of the function f is shown in the figure. Which of the following statements about f is true?

lim(x->a) f(x) = 2 and lim(x->b) f(x) does not exist

If 3x^2 + 2xy + y^2 = 2, then the value of dy/dx at x=1 is

not defined

What is lim x->1+ x/ln(x)

+∞

Use the linear approximation of f(x) = √(1+x) at a=0 to estimate √0.95.

0.9750

Which of the following are indeterminate forms? 0^0, ∞^0, 0^∞, 1^∞, ∞^∞, ∞^1

0^0, ∞^0, 1^∞

When x=8, the rate at which 3√x is increasing is 1/k times the rate at which x is increasing. What is the value of k?

12

Find the derivative of g(x) = sin^3(4x)

12sin^2(4x)cos(4x)

If g'(3) = 4 and h'(3) = -1, find f'(3) for f(x) = 5g(x) + 3h(x) + 2

17

If g'(4) = 4 and h'(4) = -1, find f'(4) for f(x) = 5g(x) + 3h(x) + 2

17

Find f'(x) for f(x) = √(4x+2)

2/(√4x+2)

Evaluate the integral ∫4_0 x/(x+9)

4 - 9ln13 + 9ln9

If the base b of a triangle is increasing at a rate of 3 inches per minute while its height is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?

A is decreasing only when b > h

Let f(x) = 6e^x find the antiderivative

F(x) = 6e^x + C

What is an antiderivative?

The opposite of a derivative

Use the formula sin(A+B) to find sin(3t) in terms of sin(t).

(-4sin^3)(t) + 3sint

Find the intervals on which the function f(x) = (7x-1)/(x^3-4x) is continuous.

(-∞,-2) U (-2,2) U (2,∞)

Find all points of f(x) = 3x^2 + 9x the graph of whose tangent lines are parallel to the line y - 33x = 0.

(4, 84)

Graph the function f(x) = x^2 + 2 in the interval [0,2] showing partition into 4 subintervals of equal length and estimating the area under the function using mid-point sum Σ4_k=1 f(c_k) △x_k

(Goes to 3, then up to 4, then up to 5)

On a circle of radius R find the x and y coordinates at time t (and angle t).

(Rcos t, Rsin t)

The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f.

(the empty graph)

Find the slope of the tangent line to the curve y=7cosx at x=π/4

-(7√2)/2

Find f'(x) for f(x) = (lnx)^x

f'(x) = (lnx)^x (ln(lnx)) + 1/lnx

Find an equation for the line tangent to given curve y = x^2-x at x=-3

y = -7x - 9

Find an equation for the line tangent to the given curve xe^x - e^(x-y) = 0 at x = 1

y = -x + 1

Find an equation for the line tangent to given curve y = (-10x^2 - 3)/(4x + 1) at x = 0

y = 12x - 3

If y^x = lnx then find dy/dx

y/(x^2 y^2) - (ylny)/x

If f(t)=t^2 + t , find (f(t+h)-f(t))/h.

2t + h + 1

A person 2 meters tall walks directly away from a streetlight that is 8 meters above the ground. If the person is walking at a constant rate and the person's shadow is lengthening at the rate of 4/9 meters per second, at what rate, in meters per second, is the person walking?

4/3

Estimate the area under the graph f(x) = x^2 between x=0 and x=4 using right sum with two rectangles of equal width.

40