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Math 111 – Calculus 1 Practice Problems for Exam 2 Fall 2008
- Find the derivatives of the following functions
(a) f (x) = x
(^2) − 3 x+
(b) y = ex(x^3 + 4x − 1)
(c) s(t) = t 22 t 2 t^5 − 1
(d) f (x) = ln(cos x)
(e) g(x) = 10sec(x)
(f) h(x) = tan−^1 (4x)
(g) s(t) = log 5 (2 − sin(t))
(h) y = sin(1 + x^2 )
(i) v = cos
− (^1) (s) 1+es
(j) z = ln(3x^2 + ln x)
(k) y = xsin^ x
- Find all values for which the tangent line to the graph of y = (^) (xx−+11) 3 is horizontal.
- For each of the following, find (^) dxdy. (Your answers will be in terms of x and y.)
(a) x^3 + (x + 1) cos(y) + y^5 − 10 = 0
(b) cos(y) = x
2 y
- Find the equation of the tangent line to y^2 + x = ex (^2) y at (0, 1).
- Show that the curves y^2 = x and 2x^2 + y^2 = 15 are orthogonal.
- A particle moves in a straight line, it’s position at time t for 0 ≤ t ≤ 2 π is given by s(t) = cos^2 t.
(a) Find the velocity of the particle at any time t.
(b) Find all times t at which the velocity of the particle is zero. What does it mean when the velocity is negative? Find all times when the velocity is positive and when velocity is negative.
(c) Find the acceleration of the particle at time t and all times at which the acceler- ation is zero. Hint: The identity sin^2 t − cos^2 t = 1 − 2 cos^2 t may be useful.
- For what values of r does the function y = erx^ satisfy the differential equation
y′′^ + 3y′^ − 4 y = 0?
- (a) A light shines from the top of a building 64 ft high. A ball is dropped from the same height, from a point 30 ft away from the light. The balls height above the ground at any time t is 50 − 32 t^2 feet. How fast is the shadow of the ball moving along the ground 1 second later? (b) A police car is driving south at 25 ft/s towards an intersection. You are driving east towards the same intersection at 40 ft/s. When the police car is 300 ft from the intersection, the police officer aims his radar gun at you and takes a reading. He does this without stopping his car. You are 400 ft from the intersection when he takes the reading. Assuming that the radar gun measures the rate of change of distance between the gun and its target, what speed (in ft/s) will the radar gun indicate? (c) A spherical balloon is leaking air at a rate of 13cm^3 /minute. At what rate is the radius decreasing when the radius is 35 cm? (Volume of a sphere= 43 πr^3 .) (d) A tank has the form of an inverted cone with height 3m and radius 2m. If water is being pumped in at a rate of 10,000 cm^3 /min how fast is the water level rising when the depth of the water is 1m? (Volume of a cone= 13 πr^2 h.)
- (a) Find the differential dy if y = 3x^2 + ex, x = 0 and dx =. 12.
(b) Let f be a function such that f (3) = 4 and the graph of its derivative is as shown on page 269 of Stewart. Use a linear approximation to estimate f (2.9). (c) Use a linear approximation or differentials to estimate the value of ln(1.03). (d) Use a linear approximation (or differentials) to estimate 3
