Chapter 1.5 Homework

1.  A curve has equation y = f(x).

(a) Write an expression for the slope of the secant line through the points P(2, f(2))  and Q(x, f(x)).

(b) Write an expression for the slope of the tangent line at P.

2   Consider the parabola y = 7x − x².

(a) Find the slope of the tangent line to the parabola at the point (1, 6).

(b) Find an equation of the tangent line in part (a).

(c) Graph the parabola and the tangent line.

3.  Find an equation of the tangent line to the graph of f at the given point.

$f\left(x\right) = \sqrt{x}, \left(25,5\right)$

4.  A ball is thrown into the air with a velocity of 37 ft/s. Its height, in feet, after t seconds is given by s(t) = 37t − 16t².  Find the velocity (in ft/s) of the ball at time t = 1 second.

5.  If a rock is thrown upward on the planet Mars with a velocity of 11 m/s,  its height above the ground (in meters) after t seconds is given by H = 11t − 1.86t².

(a)  Find the velocity (in m/s) of the rock after 2 seconds.

(b) Find the velocity (in m/s) of the rock when t = a.

(c) When (in seconds) will the rock hit the surface? (Round your answer to one decimal place.)

(d) With what velocity (in m/s) will the rock hit the surface?

6.   The graph of a function f is shown.

(a) Find the average rate of change of f on the interval [50, 70].

(b) Identify an interval on which the average rate of change of f is 0.

(c) Compute the following.

$\frac{f\left(40\right)–f\left(0\right)}{40–0}$

(d) Estimate the value of f’ (50).

(e) Is f’ (10) > f’ (30)?

7.  Sketch the graph of a function f for which f(0) = 0,  f ′(0) = 3, f ′(1) = 0,  and  f ′(2) = −1.

8. A graphing calculator is recommended.

(a) If F(x) =

, find F'(2) and use it to find an equation of the tangent line to the curve y =

at the point (2,2).

F'(2) =

y(x) =

9. Illustrate part (a) by graphing the curve and the tangent line on the same screen.